V5 - Linear independence


Example 1 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -4 & 0 & 5 & -5 & -6 & 0 \\ 1 & 4 & -3 & -1 & 4 & 0 \\ 2 & 2 & 0 & -2 & -3 & 0 \\ 2 & -6 & 1 & 5 & 3 & 0 \\ -2 & 5 & 3 & -8 & -4 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] + x_{5} \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

Example 2 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\} \)is linearly independent.

Example 3 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 4 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -4 & -6 & 7 & 0 \\ 0 & 0 & 5 & -10 & 0 \\ -6 & -1 & -6 & 6 & 0 \\ 3 & -1 & 5 & -7 & 0 \\ -4 & -6 & 5 & -14 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

Example 5 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\} \)is linearly independent.

Example 6 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

Example 7 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -5 & -4 & -3 & 0 & 11 & 0 \\ -5 & -6 & -4 & 1 & 10 & 0 \\ -3 & -5 & 2 & 5 & -10 & 0 \\ -4 & 0 & -6 & 1 & 26 & 0 \\ -1 & -5 & -1 & -6 & -5 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] + x_{5} \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

Example 8 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -3 & -3 & -1 & 8 & 0 \\ 2 & -2 & -1 & 1 & 0 \\ -3 & -1 & 0 & 5 & 0 \\ -4 & 5 & -3 & -9 & 0 \\ -5 & -6 & 2 & 19 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\} \)is linearly dependent.

Example 9 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] \right\} \)is linearly independent.

Example 10 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 4 & -5 & 4 & 5 & 2 & 0 \\ -5 & -6 & -5 & 11 & 5 & 0 \\ 5 & -5 & 5 & 1 & -6 & 0 \\ 1 & 0 & 1 & 5 & -5 & 0 \\ -1 & -5 & -1 & 2 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

Example 11 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -6 & 4 & -4 & -2 & 5 & 0 \\ 0 & -5 & -2 & 9 & 0 & 0 \\ 3 & -4 & -2 & 11 & -4 & 0 \\ 3 & -5 & 0 & 8 & 1 & 0 \\ -2 & -6 & -2 & 8 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

Example 12 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 13 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 3 \\ 0 \\ -5 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 3 \\ 0 \\ -5 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -6 \\ 1 \\ 1 \\ -1 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ -5 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 4 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -1 \\ 3 \\ 0 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 3 \\ 0 \\ -5 \end{array}\right] \right\} \)is linearly independent.

Example 14 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -6 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 3 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 0 \\ -1 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -6 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 3 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 0 \\ -1 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -3 \\ 2 \\ -2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 5 \\ -6 \\ 4 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ -4 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -1 \\ 3 \\ -1 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 0 \\ -1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -3 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -6 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 3 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 0 \\ -1 \end{array}\right] \right\} \)is linearly independent.

Example 15 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -3 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ 5 \\ 0 \\ 3 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -3 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ 5 \\ 0 \\ 3 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ 5 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 3 \\ -3 \\ 5 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ -5 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -1 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 3 \\ 5 \\ 0 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -3 \\ 5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ 5 \\ 0 \\ 3 \end{array}\right] \right\} \)is linearly independent.

Example 16 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & -4 & -6 & 12 & 5 & 0 \\ -5 & 1 & -5 & -6 & -4 & 0 \\ 5 & 3 & -6 & 13 & -2 & 0 \\ 4 & 4 & -2 & 6 & -2 & 0 \\ -6 & 1 & -4 & -9 & 4 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

Example 17 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

Example 18 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -2 & -3 & -2 & -4 & 0 \\ -3 & 3 & -2 & 11 & 0 \\ -6 & 1 & -5 & 10 & 0 \\ -6 & -6 & -5 & -4 & 0 \\ -1 & -5 & 0 & -7 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\} \)is linearly dependent.

Example 19 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & 2 & 3 & -9 & -1 & 0 \\ 5 & 5 & 10 & -20 & -4 & 0 \\ -5 & 1 & -4 & 13 & 1 & 0 \\ -3 & 5 & 2 & 7 & -1 & 0 \\ 5 & -6 & -1 & -1 & -5 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

Example 20 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} 3 & 3 & -6 & 9 & 0 \\ 0 & 0 & -4 & 0 & 0 \\ -6 & -4 & 4 & -16 & 0 \\ -5 & 4 & -2 & -6 & 0 \\ -3 & 4 & 1 & -2 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\} \)is linearly dependent.

Example 21 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -4 & -1 & 9 & 0 \\ -4 & 3 & 5 & -23 & 0 \\ -5 & -1 & 2 & -6 & 0 \\ -3 & -4 & 2 & 5 & 0 \\ -2 & 2 & 3 & -14 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\} \)is linearly dependent.

Example 22 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 0 & -4 & 2 & -2 & -2 & 0 \\ -2 & -6 & 3 & 3 & 3 & 0 \\ 0 & -4 & -6 & -1 & -10 & 0 \\ 3 & 1 & 0 & 5 & -8 & 0 \\ 2 & 0 & -6 & 2 & -12 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\} \)is linearly dependent.

Example 23 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ 1 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 2 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ -6 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -5 \\ 2 \\ 4 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ 1 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 2 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ -6 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -5 \\ 2 \\ 4 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -5 \\ 1 \\ 5 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 5 \\ -1 \\ 2 \\ 3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -6 \\ -1 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 0 \\ -5 \\ 2 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -5 \\ 1 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 2 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ -6 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -5 \\ 2 \\ 4 \end{array}\right] \right\} \)is linearly independent.

Example 24 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\} \)is linearly dependent.

Example 25 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] \right\} \)is linearly independent.

Example 26 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 27 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -5 & 15 & 26 & 0 \\ -1 & -6 & 13 & 13 & 0 \\ 1 & -1 & 1 & -5 & 0 \\ 3 & 4 & -11 & -8 & 0 \\ -5 & -2 & 9 & 9 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] + x_{4} \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\} \)is linearly dependent.

Example 28 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ -3 \\ -6 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -5 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ 4 \\ 1 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ -3 \\ -6 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -5 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ 4 \\ 1 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -6 \\ 1 \\ -6 \\ 3 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ -4 \\ -3 \\ -6 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ -5 \\ 1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ 4 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 1 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ -3 \\ -6 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -5 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ 4 \\ 1 \end{array}\right] \right\} \)is linearly independent.

Example 29 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\} \)is linearly dependent.

Example 30 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ 2 \\ 4 \\ -5 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ 2 \\ 4 \\ -5 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 2 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -6 \\ 3 \\ 2 \\ 4 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ 2 \\ 4 \\ -5 \end{array}\right] \right\} \)is linearly independent.

Example 31 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly independent.

Example 32 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -3 & -1 & -3 & -2 & 20 & 0 \\ -1 & 1 & 1 & -6 & -2 & 0 \\ 4 & 4 & 4 & 3 & -32 & 0 \\ -6 & -5 & -3 & 3 & 37 & 0 \\ 0 & -6 & 5 & 2 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 & -2 & 0 \\ 0 & 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

Example 33 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -2 & 4 & 2 & 1 & 0 \\ 5 & 3 & -5 & -15 & 0 \\ 5 & 0 & -5 & -15 & 0 \\ -4 & -4 & 4 & 15 & 0 \\ -5 & -4 & 5 & 16 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\} \)is linearly dependent.

Example 34 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] \right\} \)is linearly independent.

Example 35 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -2 & 1 & -6 & 3 & 6 & 0 \\ -6 & 0 & -6 & -1 & 12 & 0 \\ 1 & 2 & 4 & -4 & 2 & 0 \\ 4 & 3 & -2 & -4 & -2 & 0 \\ -4 & -1 & 3 & 1 & 6 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\} \)is linearly dependent.

Example 36 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} 1 & -4 & -1 & 6 & 0 \\ -6 & -1 & 0 & 14 & 0 \\ 4 & -2 & -2 & -4 & 0 \\ -4 & 4 & 2 & 0 & 0 \\ -1 & -1 & 2 & 4 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

Example 37 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -6 & 0 & -6 & 16 & 0 \\ -3 & 0 & -3 & 11 & 0 \\ 5 & 4 & 1 & -33 & 0 \\ 3 & 3 & 0 & -7 & 0 \\ 5 & 3 & 2 & -23 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\} \)is linearly dependent.

Example 38 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] \right\} \)is linearly independent.

Example 39 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ 5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 5 \\ -5 \\ -2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 5 \\ -2 \\ 1 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ 4 \\ 5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 5 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ 5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 5 \\ -5 \\ -2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 5 \\ -2 \\ 1 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ 4 \\ 5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 5 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ 5 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 5 \\ -5 \\ -2 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -2 \\ 1 \\ -4 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 5 \\ 4 \\ 5 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ 5 \\ -5 \\ 0 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ 5 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 5 \\ -5 \\ -2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 5 \\ -2 \\ 1 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ 4 \\ 5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 5 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\} \)is linearly independent.

Example 40 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -2 \\ -1 \\ 0 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -2 \\ -1 \\ 0 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 5 \\ -6 \\ -4 \\ -2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -3 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ -6 \\ -1 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -2 \\ -1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -6 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -2 \\ -1 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 41 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] \right\} \)is linearly independent.

Example 42 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\} \)is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} 2 & -2 & -2 & 7 & 0 \\ -2 & -1 & -1 & -13 & 0 \\ 5 & -1 & -1 & -10 & 0 \\ 4 & -4 & -4 & 8 & 0 \\ -3 & -5 & -5 & -2 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\} \)is linearly independent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has no nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\} \)is linearly dependent.

Example 43 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 44 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -5 & -6 & 15 & 0 \\ -2 & 3 & 5 & -9 & 0 \\ 3 & 0 & 1 & 0 & 0 \\ -2 & -1 & 2 & 3 & 0 \\ 5 & -4 & 5 & 12 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\} \)is linearly dependent.

Example 45 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] \right\} \)is linearly independent.

Example 46 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\} \)is linearly dependent.

Example 47 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] \right\} \)is linearly independent.

Example 48 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\} \)is linearly independent.

Example 49 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

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

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] \right\} \)is linearly independent.

Example 50 πŸ”—

Consider the statement

The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\} \)is linearly dependent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} 5 & -1 & 0 & -16 & 0 \\ -2 & -2 & -3 & 1 & 0 \\ 3 & 0 & -1 & -10 & 0 \\ -2 & 1 & 5 & 12 & 0 \\ 3 & 5 & -3 & -7 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\} \)is linearly dependent.

    is equivalent to the statement

    The vector equation \( x_{1} \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \)has (infinitely many) nontrivial solutions.

  2. The set of vectors \( \left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\} \)is linearly dependent.