V3 - Spanning sets


Example 1 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ 2 \\ 1 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -1 & 4 & -9 & -9 & -4 \\ -1 & -3 & 12 & 12 & 4 \\ 0 & -4 & 12 & 12 & 2 \\ 3 & -4 & 3 & 3 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & -3 & 0 \\ 0 & 1 & -3 & -3 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ 2 \\ 1 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} -4 \\ 4 \\ 2 \\ 1 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ 12 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ 2 \\ 1 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 2 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 15 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -15 \\ 2 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -3 \\ -2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -5 & -5 & 15 & -15 & 0 \\ 2 & 0 & -2 & 2 & 2 \\ -1 & -1 & 3 & -3 & -3 \\ 3 & 1 & -5 & 5 & -2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 \\ 0 & 1 & -2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 15 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -15 \\ 2 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -3 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 2 \\ -1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 15 \\ -2 \\ 3 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -15 \\ 2 \\ -3 \\ 5 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 2 \\ -3 \\ -2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 15 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -15 \\ 2 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -3 \\ -2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 3 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 4 \\ 9 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & 4 & 3 & 0 & 4 & 1 \\ -2 & 3 & 3 & -5 & -2 & -3 \\ -1 & -5 & -5 & 4 & 2 & 3 \\ 4 & 1 & -2 & 9 & 4 & 0 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 1 & 0 & \frac{2}{9} \\ 0 & 1 & 0 & 1 & 0 & -\frac{14}{9} \\ 0 & 0 & 1 & -2 & 0 & \frac{11}{9} \\ 0 & 0 & 0 & 0 & 1 & \frac{7}{9} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 4 \\ 9 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ -5 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 3 \\ -5 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -5 \\ 4 \\ 9 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ -2 \\ 2 \\ 4 \end{array}\right] + x_{6} \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 4 \\ 9 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 4 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 2 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & -1 & -5 & 2 \\ -4 & -4 & 2 & 1 \\ -2 & 3 & 4 & -2 \\ -2 & 2 & -1 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ -4 \\ -2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -4 \\ 3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 2 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 5 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -12 \\ 2 \\ 12 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -2 \\ -5 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -2 & -5 & 1 & -12 & 0 & -3 \\ -2 & -1 & -2 & 2 & 0 & -4 \\ -3 & 1 & -5 & 12 & -1 & -2 \\ -2 & 2 & 3 & -2 & -3 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & \frac{285}{61} \\ 0 & 1 & 0 & 2 & 0 & -\frac{100}{61} \\ 0 & 0 & 1 & -2 & 0 & -\frac{113}{61} \\ 0 & 0 & 0 & 0 & 1 & -\frac{268}{61} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -12 \\ 2 \\ 12 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -2 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -2 \\ -5 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -12 \\ 2 \\ 12 \\ -2 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} -3 \\ -4 \\ -2 \\ -5 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -12 \\ 2 \\ 12 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -2 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 6 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -18 \\ 0 \\ 18 \\ 21 \end{array}\right] , \left[\begin{array}{c} 40 \\ 8 \\ -46 \\ -50 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & 4 & -18 & 40 & 3 & 2 \\ 4 & -4 & 0 & 8 & -3 & 0 \\ -5 & -1 & 18 & -46 & 1 & -5 \\ -4 & -3 & 21 & -50 & 4 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -3 & 8 & 0 & 0 \\ 0 & 1 & -3 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -18 \\ 0 \\ 18 \\ 21 \end{array}\right] , \left[\begin{array}{c} 40 \\ 8 \\ -46 \\ -50 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ 4 \\ -5 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -4 \\ -1 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} -18 \\ 0 \\ 18 \\ 21 \end{array}\right] + x_{4} \left[\begin{array}{c} 40 \\ 8 \\ -46 \\ -50 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ -3 \\ 1 \\ 4 \end{array}\right] + x_{6} \left[\begin{array}{c} 2 \\ 0 \\ -5 \\ -2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -18 \\ 0 \\ 18 \\ 21 \end{array}\right] , \left[\begin{array}{c} 40 \\ 8 \\ -46 \\ -50 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 7 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -5 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 4 & -5 & 3 & 4 & 1 & -3 \\ 0 & -3 & -3 & 0 & -2 & 4 \\ 1 & -1 & 3 & 3 & -1 & -5 \\ -1 & -2 & -1 & 2 & -2 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & \frac{344}{61} \\ 0 & 1 & 0 & -1 & 0 & \frac{177}{61} \\ 0 & 0 & 1 & 1 & 0 & -\frac{191}{61} \\ 0 & 0 & 0 & 0 & 1 & -\frac{101}{61} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -5 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ -2 \end{array}\right] + x_{6} \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -5 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 8 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -1 \\ -1 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & 3 & -5 & -3 & 1 & 2 \\ 4 & -2 & -5 & 1 & 4 & 1 \\ 2 & -5 & 3 & 4 & 0 & -1 \\ -5 & -5 & 4 & 3 & -4 & -1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{2}{37} & \frac{7}{37} \\ 0 & 1 & 0 & 0 & \frac{77}{37} & -\frac{29}{37} \\ 0 & 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & \frac{125}{37} & -\frac{49}{37} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -1 \\ -1 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ 4 \\ 2 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} 1 \\ 4 \\ 0 \\ -4 \end{array}\right] + x_{6} \left[\begin{array}{c} 2 \\ 1 \\ -1 \\ -1 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -1 \\ -1 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 9 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -1 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 1 \\ 4 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & -3 & -4 & -4 & -1 & -1 \\ -1 & 4 & 2 & -3 & -1 & -3 \\ -3 & 4 & -5 & -4 & 1 & 1 \\ 4 & 1 & -3 & 1 & -3 & 4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{43}{55} & \frac{53}{330} \\ 0 & 1 & 0 & 0 & -\frac{47}{165} & \frac{146}{495} \\ 0 & 0 & 1 & 0 & -\frac{14}{165} & -\frac{731}{990} \\ 0 & 0 & 0 & 1 & \frac{26}{165} & \frac{839}{990} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -1 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 1 \\ 4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -1 \\ -3 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 4 \\ 4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 2 \\ -5 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -3 \\ -4 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} -1 \\ -3 \\ 1 \\ 4 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -1 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 1 \\ 4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 10 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -5 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 3 \\ -2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -2 & -2 & -2 & -4 & -3 & 4 \\ -5 & -1 & 3 & 3 & 4 & -1 \\ 2 & -5 & -2 & -5 & -5 & 3 \\ -2 & 2 & 4 & -2 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{3}{10} & -\frac{24}{55} \\ 0 & 1 & 0 & 0 & -\frac{1}{10} & -\frac{8}{55} \\ 0 & 0 & 1 & 0 & -\frac{3}{10} & -\frac{4}{5} \\ 0 & 0 & 0 & 1 & \frac{11}{10} & -\frac{17}{55} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -5 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 3 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ -5 \\ 2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -1 \\ -5 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 3 \\ -5 \\ -2 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} 4 \\ -1 \\ 3 \\ -2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -5 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 3 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 11 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 10 \\ 0 \\ 8 \\ 12 \end{array}\right] , \left[\begin{array}{c} 12 \\ 9 \\ -3 \\ 9 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & -2 & 10 & 12 \\ 4 & 1 & 0 & 9 \\ -4 & -3 & 8 & -3 \\ 0 & -3 & 12 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 3 \\ 0 & 1 & -4 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 10 \\ 0 \\ 8 \\ 12 \end{array}\right] , \left[\begin{array}{c} 12 \\ 9 \\ -3 \\ 9 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ 4 \\ -4 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 10 \\ 0 \\ 8 \\ 12 \end{array}\right] + x_{4} \left[\begin{array}{c} 12 \\ 9 \\ -3 \\ 9 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 10 \\ 0 \\ 8 \\ 12 \end{array}\right] , \left[\begin{array}{c} 12 \\ 9 \\ -3 \\ 9 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 12 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -3 \\ 3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -3 & 2 & -1 & -3 & -4 \\ -4 & 1 & 0 & -5 & -4 \\ -5 & 1 & 1 & 0 & -3 \\ 2 & -1 & -1 & 1 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{1}{8} \\ 0 & 1 & 0 & 0 & -\frac{13}{8} \\ 0 & 0 & 1 & 0 & -\frac{3}{4} \\ 0 & 0 & 0 & 1 & \frac{3}{8} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -3 \\ 3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -5 \\ 0 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -4 \\ -4 \\ -3 \\ 3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -3 \\ 3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 13 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ -3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 4 & -5 & 1 & -2 & 0 \\ 1 & -5 & 1 & 0 & 1 \\ 1 & -2 & 3 & -3 & -1 \\ -5 & 1 & 0 & -5 & -3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{1}{39} \\ 0 & 1 & 0 & 0 & -\frac{7}{39} \\ 0 & 0 & 1 & 0 & \frac{1}{13} \\ 0 & 0 & 0 & 1 & \frac{7}{13} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ -3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ -5 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ -3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ -3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 14 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 14 \\ -1 \\ 1 \\ 18 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & -4 & 0 & 14 & -5 & -2 \\ -4 & -3 & 2 & -1 & 1 & -4 \\ 1 & -2 & 2 & 1 & 2 & -4 \\ 3 & -5 & 0 & 18 & -1 & -4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 1 & 0 & -\frac{3}{29} \\ 0 & 1 & 0 & -3 & 0 & \frac{23}{29} \\ 0 & 0 & 1 & -3 & 0 & -\frac{51}{58} \\ 0 & 0 & 0 & 0 & 1 & -\frac{8}{29} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 14 \\ -1 \\ 1 \\ 18 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -3 \\ -2 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 2 \\ 2 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 14 \\ -1 \\ 1 \\ 18 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 1 \\ 2 \\ -1 \end{array}\right] + x_{6} \left[\begin{array}{c} -2 \\ -4 \\ -4 \\ -4 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 14 \\ -1 \\ 1 \\ 18 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 15 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 3 \\ -3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -1 & 1 & 4 & 4 & 0 \\ -1 & -2 & 2 & -3 & 3 \\ -4 & 2 & -1 & 4 & 3 \\ 3 & 0 & -5 & -4 & -3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & -\frac{26}{21} \\ 0 & 1 & 0 & 0 & -\frac{10}{21} \\ 0 & 0 & 1 & 0 & \frac{1}{21} \\ 0 & 0 & 0 & 1 & -\frac{5}{21} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 3 \\ -3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ 2 \\ -1 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ -3 \\ 4 \\ -4 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 3 \\ 3 \\ -3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 3 \\ -3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 16 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -13 \\ -15 \\ 17 \\ 8 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 3 & -1 & -13 \\ 3 & 4 & 2 & -15 \\ -2 & -5 & -5 & 17 \\ 4 & -4 & 1 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -13 \\ -15 \\ 17 \\ 8 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 3 \\ -2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -13 \\ -15 \\ 17 \\ 8 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -13 \\ -15 \\ 17 \\ 8 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 17 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -3 \\ 4 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -5 & -5 & 20 & 20 & 2 \\ -3 & 2 & 2 & 2 & 1 \\ -1 & -3 & 8 & 8 & -3 \\ -3 & -2 & 10 & 10 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & -2 & 0 \\ 0 & 1 & -2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -3 \\ 4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 2 \\ -3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] + x_{4} \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 1 \\ -3 \\ 4 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 20 \\ 2 \\ 8 \\ 10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -3 \\ 4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 18 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -5 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 4 & -2 & 0 & 0 & -1 & -4 \\ 1 & 3 & 1 & 4 & 0 & 3 \\ 2 & -2 & 1 & -1 & -1 & 3 \\ 4 & -2 & 1 & 2 & 0 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{19}{34} & \frac{10}{17} \\ 0 & 1 & 0 & 0 & -\frac{21}{34} & \frac{54}{17} \\ 0 & 0 & 1 & 0 & -\frac{7}{17} & \frac{87}{17} \\ 0 & 0 & 0 & 1 & \frac{12}{17} & -\frac{52}{17} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -5 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 1 \\ 2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ 0 \\ -1 \\ 0 \end{array}\right] + x_{6} \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -5 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 19 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ -9 \\ -9 \end{array}\right] , \left[\begin{array}{c} -8 \\ 10 \\ -9 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ -2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 2 & -3 & -9 & -8 & -5 & -1 \\ -2 & 4 & 12 & 10 & -2 & 0 \\ 3 & -3 & -9 & -9 & 0 & 0 \\ -5 & -3 & -9 & -1 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 3 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ -9 \\ -9 \end{array}\right] , \left[\begin{array}{c} -8 \\ 10 \\ -9 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -2 \\ 3 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} -9 \\ 12 \\ -9 \\ -9 \end{array}\right] + x_{4} \left[\begin{array}{c} -8 \\ 10 \\ -9 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ -2 \\ 0 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ -2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -9 \\ 12 \\ -9 \\ -9 \end{array}\right] , \left[\begin{array}{c} -8 \\ 10 \\ -9 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 20 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ -5 \\ -4 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 0 & 2 & -4 & -4 & -1 & 3 \\ 4 & 3 & -3 & 2 & -3 & -4 \\ -5 & 4 & 3 & -1 & 4 & -5 \\ 2 & -1 & 2 & -1 & 1 & -4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{70}{311} & -\frac{349}{311} \\ 0 & 1 & 0 & 0 & \frac{97}{622} & -\frac{907}{622} \\ 0 & 0 & 1 & 0 & \frac{401}{622} & -\frac{973}{622} \\ 0 & 0 & 0 & 1 & -\frac{197}{622} & \frac{53}{622} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ -5 \\ -4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 0 \\ 4 \\ -5 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -3 \\ 3 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ 1 \end{array}\right] + x_{6} \left[\begin{array}{c} 3 \\ -4 \\ -5 \\ -4 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 4 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ -5 \\ -4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 21 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 1 & -5 & -1 & 4 & -4 & 2 \\ -5 & 4 & -5 & 3 & -3 & -5 \\ -1 & -5 & 1 & 0 & -3 & 3 \\ -5 & -2 & 3 & 4 & -5 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{579}{1028} & \frac{73}{1028} \\ 0 & 1 & 0 & 0 & \frac{271}{514} & -\frac{265}{514} \\ 0 & 0 & 1 & 0 & \frac{205}{1028} & \frac{507}{1028} \\ 0 & 0 & 0 & 1 & -\frac{111}{257} & -\frac{10}{257} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ -5 \end{array}\right] + x_{6} \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 22 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 4 \\ 3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -2 & 1 & -3 & 4 \\ 0 & -3 & 3 & -3 & 0 \\ 2 & 0 & -3 & -3 & 4 \\ 0 & 3 & -1 & 5 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 4 \\ 3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -3 \\ -3 \\ 5 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ 0 \\ 4 \\ 3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 4 \\ 3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 23 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 1 \\ 2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -3 & -2 & 4 & -3 & -3 \\ 0 & 2 & -1 & 4 & 1 \\ -1 & -4 & -4 & 2 & 1 \\ -3 & 4 & -3 & 3 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{19}{121} \\ 0 & 1 & 0 & 0 & \frac{5}{22} \\ 0 & 0 & 1 & 0 & -\frac{62}{121} \\ 0 & 0 & 0 & 1 & \frac{1}{121} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 4 \\ 2 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 1 \\ 1 \\ 2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 24 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 3 & 1 & -3 & -5 & 3 & -2 \\ 2 & -1 & 3 & -1 & 4 & -3 \\ -4 & 2 & -3 & 2 & -4 & -4 \\ 4 & 4 & 0 & 1 & -2 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{11}{19} & \frac{51}{19} \\ 0 & 1 & 0 & 0 & \frac{28}{57} & -\frac{268}{57} \\ 0 & 0 & 1 & 0 & \frac{4}{3} & -\frac{10}{3} \\ 0 & 0 & 0 & 1 & -\frac{94}{57} & \frac{175}{57} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -1 \\ 2 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 3 \\ -3 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ -1 \\ 2 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 4 \\ -4 \\ -2 \end{array}\right] + x_{6} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 25 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 16 \\ -7 \\ -4 \end{array}\right] , \left[\begin{array}{c} -17 \\ 34 \\ -23 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 1 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -5 & 2 & -3 & -17 & 3 \\ 4 & -5 & 16 & 34 & 1 \\ -5 & 3 & -7 & -23 & 2 \\ 4 & 0 & -4 & 4 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 1 & 0 \\ 0 & 1 & -4 & -6 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 16 \\ -7 \\ -4 \end{array}\right] , \left[\begin{array}{c} -17 \\ 34 \\ -23 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 1 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 16 \\ -7 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} -17 \\ 34 \\ -23 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 1 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 16 \\ -7 \\ -4 \end{array}\right] , \left[\begin{array}{c} -17 \\ 34 \\ -23 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 1 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 26 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 0 \\ -3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & -3 & -1 & 0 \\ -3 & 2 & 1 & -3 \\ 4 & 4 & 4 & 0 \\ -5 & 1 & 2 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 0 \\ -3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -3 \\ 4 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 2 \\ 4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -3 \\ 0 \\ -3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 0 \\ -3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 27 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 6 \\ 13 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 1 & 4 & 5 \\ 1 & 1 & 4 & -1 \\ -1 & 3 & -5 & 6 \\ 3 & 4 & -3 & 13 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 6 \\ 13 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ 4 \\ -5 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ -1 \\ 6 \\ 13 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 6 \\ 13 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 28 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 8 \\ 3 \\ 0 \\ 20 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -3 \\ -3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 4 & -1 & -2 & 8 & 0 & -5 \\ 3 & 2 & 1 & 3 & 4 & -4 \\ 0 & 1 & 2 & 0 & -1 & -3 \\ 4 & -4 & 4 & 20 & 2 & -3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 2 & 0 & -\frac{183}{104} \\ 0 & 1 & 0 & -2 & 0 & -\frac{161}{156} \\ 0 & 0 & 1 & 1 & 0 & -\frac{157}{312} \\ 0 & 0 & 0 & 0 & 1 & \frac{25}{26} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 8 \\ 3 \\ 0 \\ 20 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -3 \\ -3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 8 \\ 3 \\ 0 \\ 20 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] + x_{6} \left[\begin{array}{c} -5 \\ -4 \\ -3 \\ -3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 8 \\ 3 \\ 0 \\ 20 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -3 \\ -3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 29 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 4 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -1 & 0 & -1 & 1 & -5 & 1 \\ -2 & 3 & -3 & -6 & 2 & 0 \\ -4 & 2 & -5 & -2 & -4 & 0 \\ 0 & 2 & 3 & 2 & 4 & 4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -3 & 0 & -\frac{5}{7} \\ 0 & 1 & 0 & -2 & 0 & \frac{6}{7} \\ 0 & 0 & 1 & 2 & 0 & \frac{8}{7} \\ 0 & 0 & 0 & 0 & 1 & -\frac{2}{7} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 3 \\ 2 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{6} \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 4 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 30 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 15 \\ -12 \\ 15 \\ 15 \end{array}\right] , \left[\begin{array}{c} -6 \\ 10 \\ -2 \\ -10 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -5 & -2 & 15 & -6 \\ 4 & -1 & -12 & 10 \\ -5 & -4 & 15 & -2 \\ -5 & 0 & 15 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 15 \\ -12 \\ 15 \\ 15 \end{array}\right] , \left[\begin{array}{c} -6 \\ 10 \\ -2 \\ -10 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -1 \\ -4 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 15 \\ -12 \\ 15 \\ 15 \end{array}\right] + x_{4} \left[\begin{array}{c} -6 \\ 10 \\ -2 \\ -10 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 15 \\ -12 \\ 15 \\ 15 \end{array}\right] , \left[\begin{array}{c} -6 \\ 10 \\ -2 \\ -10 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 31 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 8 \\ 10 \\ -8 \\ -16 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ -4 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 3 & 8 & 0 & 3 \\ -1 & 3 & 10 & 0 & 2 \\ -4 & -4 & -8 & 0 & 4 \\ 1 & -5 & -16 & 0 & -4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 8 \\ 10 \\ -8 \\ -16 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ -4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 8 \\ 10 \\ -8 \\ -16 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ -4 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 8 \\ 10 \\ -8 \\ -16 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ -4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 32 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 3 & -2 & -1 & -4 & -2 & 1 \\ -3 & 4 & 4 & -5 & 2 & -3 \\ 0 & -2 & -2 & -4 & -3 & -4 \\ 0 & 1 & 3 & -5 & 3 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{43}{114} & \frac{48}{19} \\ 0 & 1 & 0 & 0 & -\frac{27}{19} & \frac{77}{19} \\ 0 & 0 & 1 & 0 & \frac{81}{38} & -\frac{49}{19} \\ 0 & 0 & 0 & 1 & \frac{15}{38} & \frac{5}{19} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 4 \\ -2 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -5 \\ -4 \\ -5 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ 2 \\ -3 \\ 3 \end{array}\right] + x_{6} \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ -5 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 33 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -18 \\ -4 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -4 & 2 & 5 \\ 4 & -4 & 0 & -4 \\ 4 & 3 & 0 & -18 \\ 1 & -1 & 1 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -18 \\ -4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ 4 \\ 4 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -4 \\ 3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ -4 \\ -18 \\ -4 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 4 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -18 \\ -4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 34 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 14 \\ -2 \\ -1 \\ -20 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 3 \\ 1 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 0 & 2 & -5 & 14 & -3 \\ 0 & 3 & 4 & -2 & 3 \\ -1 & -3 & -1 & -1 & 3 \\ 4 & -1 & 3 & -20 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 14 \\ -2 \\ -1 \\ -20 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 3 \\ 1 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 14 \\ -2 \\ -1 \\ -20 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 3 \\ 3 \\ 1 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 14 \\ -2 \\ -1 \\ -20 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 3 \\ 1 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 35 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -14 \\ 4 \\ 9 \\ -9 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -4 & -2 & 4 & -14 & 2 & -5 \\ -3 & -1 & -4 & 4 & 0 & -2 \\ 0 & 3 & -3 & 9 & 3 & -2 \\ -1 & -4 & 2 & -9 & 3 & 0 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 1 & 0 & \frac{19}{16} \\ 0 & 1 & 0 & 1 & 0 & -\frac{329}{528} \\ 0 & 0 & 1 & -2 & 0 & -\frac{31}{132} \\ 0 & 0 & 0 & 0 & 1 & -\frac{49}{176} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -14 \\ 4 \\ 9 \\ -9 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -1 \\ 3 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -4 \\ -3 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -14 \\ 4 \\ 9 \\ -9 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ 3 \end{array}\right] + x_{6} \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 0 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -14 \\ 4 \\ 9 \\ -9 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 36 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 10 \\ -10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 2 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -5 & 0 & -5 & 0 & 2 \\ -3 & 0 & -2 & 2 & 2 \\ -5 & -3 & 3 & 10 & -1 \\ 0 & -2 & -3 & -10 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 10 \\ -10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 0 \\ -3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 2 \\ 10 \\ -10 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 2 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 10 \\ -10 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 37 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 20 \\ 8 \\ -16 \\ 16 \end{array}\right] , \left[\begin{array}{c} -55 \\ -18 \\ 41 \\ -45 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -5 & -5 & 20 & -55 \\ 2 & -2 & 8 & -18 \\ 1 & 4 & -16 & 41 \\ -5 & -4 & 16 & -45 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & -4 & 10 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 20 \\ 8 \\ -16 \\ 16 \end{array}\right] , \left[\begin{array}{c} -55 \\ -18 \\ 41 \\ -45 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -2 \\ 4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 20 \\ 8 \\ -16 \\ 16 \end{array}\right] + x_{4} \left[\begin{array}{c} -55 \\ -18 \\ 41 \\ -45 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 20 \\ 8 \\ -16 \\ 16 \end{array}\right] , \left[\begin{array}{c} -55 \\ -18 \\ 41 \\ -45 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 38 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -1 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -5 & -4 & -2 & 4 \\ 4 & 2 & 3 & 3 \\ -2 & -4 & 2 & 0 \\ 3 & 4 & -2 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -1 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -1 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -1 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 39 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 5 \\ -11 \\ -2 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & -3 & 1 & -7 \\ -2 & 3 & -3 & 5 \\ 2 & -4 & -1 & -11 \\ 1 & 1 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 5 \\ -11 \\ -2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -2 \\ 2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 3 \\ -4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -3 \\ -1 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -7 \\ 5 \\ -11 \\ -2 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 5 \\ -11 \\ -2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 40 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} -11 \\ 10 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -17 \\ 20 \\ 1 \\ -12 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & -2 & -11 & -17 \\ -5 & -5 & 10 & 20 \\ -1 & -4 & -1 & 1 \\ 3 & 3 & -6 & -12 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & -5 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} -11 \\ 10 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -17 \\ 20 \\ 1 \\ -12 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ -5 \\ -1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} -11 \\ 10 \\ -1 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -17 \\ 20 \\ 1 \\ -12 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} -11 \\ 10 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -17 \\ 20 \\ 1 \\ -12 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 41 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 1 \\ -5 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 2 & -1 & -5 \\ -3 & 3 & 4 & 3 \\ -5 & -3 & -2 & 1 \\ -2 & -1 & 4 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 1 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 3 \\ 1 \\ -5 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 1 \\ -5 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 42 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} 6 \\ 1 \\ -10 \\ -9 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ 3 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 3 & 6 & 0 & -5 \\ 1 & 0 & 1 & 0 & 2 \\ -5 & -5 & -10 & 0 & 4 \\ -4 & -5 & -9 & 0 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} 6 \\ 1 \\ -10 \\ -9 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ 3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ 1 \\ -5 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 0 \\ -5 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 6 \\ 1 \\ -10 \\ -9 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ 3 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} 6 \\ 1 \\ -10 \\ -9 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 4 \\ 3 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

Example 43 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ -3 \\ 2 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -4 & -2 & -3 & -4 & -1 & -5 \\ -4 & 4 & -2 & -4 & -1 & 3 \\ 4 & -5 & 2 & 0 & -5 & -3 \\ -3 & 1 & -5 & 2 & 1 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{373}{142} & \frac{269}{142} \\ 0 & 1 & 0 & 0 & -\frac{23}{71} & \frac{111}{71} \\ 0 & 0 & 1 & 0 & \frac{138}{71} & -\frac{98}{71} \\ 0 & 0 & 0 & 1 & \frac{449}{284} & -\frac{111}{284} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ -3 \\ 2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 4 \\ -5 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -2 \\ 2 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ -1 \\ -5 \\ 1 \end{array}\right] + x_{6} \left[\begin{array}{c} -5 \\ 3 \\ -3 \\ 2 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ -3 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 44 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 2 \\ 3 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 1 & 0 & 3 & 2 & 4 \\ -2 & 3 & -1 & 1 & -3 & -4 \\ -2 & 2 & -3 & 0 & -5 & 2 \\ -4 & 1 & 0 & -1 & -2 & 3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & -\frac{75}{43} \\ 0 & 1 & 0 & 0 & -1 & -\frac{283}{86} \\ 0 & 0 & 1 & 0 & 1 & -\frac{73}{43} \\ 0 & 0 & 0 & 1 & 1 & \frac{59}{86} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 2 \\ 3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ -2 \\ -2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -1 \\ -3 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] + x_{6} \left[\begin{array}{c} 4 \\ -4 \\ 2 \\ 3 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -3 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 2 \\ 3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 45 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & -5 & 2 & 3 \\ 2 & -3 & 4 & 4 \\ 1 & -4 & 1 & -1 \\ -3 & 1 & -5 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 2 \\ 1 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -3 \\ -4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 4 \\ 1 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 4 \\ -1 \\ 2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 46 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -6 \\ -12 \end{array}\right] , \left[\begin{array}{c} 9 \\ -8 \\ 4 \\ 17 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -3 & -2 & -6 & 9 & -3 & -2 \\ 2 & 2 & 6 & -8 & 3 & -1 \\ 2 & -2 & -6 & 4 & 2 & 4 \\ -5 & -4 & -12 & 17 & 2 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 3 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -6 \\ -12 \end{array}\right] , \left[\begin{array}{c} 9 \\ -8 \\ 4 \\ 17 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 6 \\ -6 \\ -12 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ -8 \\ 4 \\ 17 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 3 \\ 2 \\ 2 \end{array}\right] + x_{6} \left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -6 \\ -12 \end{array}\right] , \left[\begin{array}{c} 9 \\ -8 \\ 4 \\ 17 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 47 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -5 & -3 & -2 & 0 & -4 & 1 \\ -1 & -4 & 2 & -2 & 1 & -4 \\ -4 & 0 & -2 & 4 & 3 & -4 \\ -1 & -5 & -5 & -2 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{239}{176} & -\frac{13}{8} \\ 0 & 1 & 0 & 0 & -\frac{255}{176} & \frac{21}{8} \\ 0 & 0 & 1 & 0 & \frac{137}{176} & -\frac{3}{8} \\ 0 & 0 & 0 & 1 & \frac{879}{352} & -\frac{45}{16} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -2 \\ 4 \\ -2 \end{array}\right] + x_{5} \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} 1 \\ -4 \\ -4 \\ -4 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ -4 \\ -4 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 48 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ -2 \end{array}\right] \right\} \)does not span \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -2 & -3 & -2 & 0 & -1 & -1 \\ 1 & 1 & -3 & -5 & 2 & -1 \\ 4 & 0 & 0 & -2 & -3 & -4 \\ -3 & 4 & 1 & -1 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{77}{36} & \frac{29}{18} \\ 0 & 1 & 0 & 0 & \frac{13}{3} & \frac{11}{3} \\ 0 & 0 & 1 & 0 & -\frac{293}{36} & -\frac{119}{18} \\ 0 & 0 & 0 & 1 & \frac{52}{9} & \frac{47}{9} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ -2 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 1 \\ 4 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ 2 \\ -3 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ -2 \end{array}\right] =\vec{v}\) is inconsistent for some vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -4 \\ -2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 49 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -3 & -2 & 4 & -3 \\ -3 & -5 & -4 & 4 & 4 \\ -4 & 4 & 4 & -2 & -1 \\ 4 & -5 & -3 & -1 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & -\frac{47}{54} \\ 0 & 1 & 0 & 0 & \frac{29}{54} \\ 0 & 0 & 1 & 0 & -\frac{62}{27} \\ 0 & 0 & 0 & 1 & -\frac{23}{18} \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -5 \\ 4 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 4 \\ -2 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 4 \\ -1 \\ 2 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

Example 50 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ 6 \end{array}\right] , \left[\begin{array}{c} 9 \\ -6 \\ 6 \\ 6 \end{array}\right] \right\} \)spans \(\mathbb{R}^4\).

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 3 & -3 & 9 \\ -3 & -2 & -4 & -6 \\ -3 & 2 & -8 & 6 \\ 4 & 2 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 2 & 0 \\ 0 & 1 & -1 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ 6 \end{array}\right] , \left[\begin{array}{c} 9 \\ -6 \\ 6 \\ 6 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\).

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 0 \\ -3 \\ -3 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ 6 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ -6 \\ 6 \\ 6 \end{array}\right] =\vec{v}\) has a solution for every vector \(\vec{v}\) in \(\mathbb{R}^4\).

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ 6 \end{array}\right] , \left[\begin{array}{c} 9 \\ -6 \\ 6 \\ 6 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\).