V8 - Polynomial and matrix spaces


Example 1 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & -1 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} 2 & -5 \\ 1 & -1 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 1 & -4 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ 4 & -3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -3 & -3 & 2 & 0 & -2 \\ -1 & -4 & -5 & -1 & 3 \\ 2 & -3 & 1 & 1 & 4 \\ 1 & 1 & -1 & -4 & -3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{29}{32} \\ 0 & 1 & 0 & 0 & -\frac{241}{416} \\ 0 & 0 & 1 & 0 & -\frac{53}{104} \\ 0 & 0 & 0 & 1 & \frac{399}{416} \end{array}\right] \]

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & -1 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} 2 & -5 \\ 1 & -1 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 1 & -4 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ 4 & -3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -3 & -1 \\ 2 & 1 \end{array}\right] + y_{2} \left[\begin{array}{cc} -3 & -4 \\ -3 & 1 \end{array}\right] + y_{3} \left[\begin{array}{cc} 2 & -5 \\ 1 & -1 \end{array}\right] + y_{4} \left[\begin{array}{cc} 0 & -1 \\ 1 & -4 \end{array}\right] + y_{5} \left[\begin{array}{cc} -2 & 3 \\ 4 & -3 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & -1 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} 2 & -5 \\ 1 & -1 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 1 & -4 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ 4 & -3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 2 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -5 \\ -4 & 2 \end{array}\right] , \left[\begin{array}{cc} 0 & -3 \\ -1 & -4 \end{array}\right] , \left[\begin{array}{cc} -1 & 4 \\ -2 & -4 \end{array}\right] , \left[\begin{array}{cc} -4 & -1 \\ -2 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -5 \\ -4 & 2 \end{array}\right] , \left[\begin{array}{cc} 0 & -3 \\ -1 & -4 \end{array}\right] , \left[\begin{array}{cc} -1 & 4 \\ -2 & -4 \end{array}\right] , \left[\begin{array}{cc} -4 & -1 \\ -2 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 2 & -5 \\ -4 & 2 \end{array}\right] + y_{2} \left[\begin{array}{cc} 0 & -3 \\ -1 & -4 \end{array}\right] + y_{3} \left[\begin{array}{cc} -1 & 4 \\ -2 & -4 \end{array}\right] + y_{4} \left[\begin{array}{cc} -4 & -1 \\ -2 & 3 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -5 \\ -4 & 2 \end{array}\right] , \left[\begin{array}{cc} 0 & -3 \\ -1 & -4 \end{array}\right] , \left[\begin{array}{cc} -1 & 4 \\ -2 & -4 \end{array}\right] , \left[\begin{array}{cc} -4 & -1 \\ -2 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 3 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 0 \\ 2 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & 0 \\ -3 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 0 \\ -8 & 13 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 14 & -21 \end{array}\right] , \left[\begin{array}{cc} -2 & -2 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -5 & 3 \\ 0 & -2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 0 & 3 & -3 & -2 & -5 \\ 0 & 0 & 0 & 0 & -2 & 3 \\ 2 & -3 & -8 & 14 & 2 & 0 \\ -5 & 4 & 13 & -21 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & 2 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 0 \\ 2 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & 0 \\ -3 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 0 \\ -8 & 13 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 14 & -21 \end{array}\right] , \left[\begin{array}{cc} -2 & -2 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -5 & 3 \\ 0 & -2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -3 & 0 \\ 2 & -5 \end{array}\right] + y_{2} \left[\begin{array}{cc} 0 & 0 \\ -3 & 4 \end{array}\right] + y_{3} \left[\begin{array}{cc} 3 & 0 \\ -8 & 13 \end{array}\right] + y_{4} \left[\begin{array}{cc} -3 & 0 \\ 14 & -21 \end{array}\right] + y_{5} \left[\begin{array}{cc} -2 & -2 \\ 2 & 1 \end{array}\right] + y_{6} \left[\begin{array}{cc} -5 & 3 \\ 0 & -2 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 0 \\ 2 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & 0 \\ -3 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 0 \\ -8 & 13 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 14 & -21 \end{array}\right] , \left[\begin{array}{cc} -2 & -2 \\ 2 & 1 \end{array}\right] , \left[\begin{array}{cc} -5 & 3 \\ 0 & -2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 4 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -2 & -2 \\ 3 & -1 \end{array}\right] + y_{2} \left[\begin{array}{cc} 4 & -5 \\ 0 & -1 \end{array}\right] + y_{3} \left[\begin{array}{cc} 0 & -1 \\ 1 & -2 \end{array}\right] + y_{4} \left[\begin{array}{cc} -5 & -1 \\ -2 & -1 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 5 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -3 \, x^{3} - 2 \, x^{2} + 3 \, x + 2 , -4 \, x^{3} - 3 \, x^{2} - x , -5 \, x^{3} + 2 \, x^{2} - 1 , 5 \, x^{3} + 1 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -3 \, x^{3} - 2 \, x^{2} + 3 \, x + 2 , -4 \, x^{3} - 3 \, x^{2} - x , -5 \, x^{3} + 2 \, x^{2} - 1 , 5 \, x^{3} + 1 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -3 \, x^{3} - 2 \, x^{2} + 3 \, x + 2 \right) + y_{2} \left( -4 \, x^{3} - 3 \, x^{2} - x \right) + y_{3} \left( -5 \, x^{3} + 2 \, x^{2} - 1 \right) + y_{4} \left( 5 \, x^{3} + 1 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -3 \, x^{3} - 2 \, x^{2} + 3 \, x + 2 , -4 \, x^{3} - 3 \, x^{2} - x , -5 \, x^{3} + 2 \, x^{2} - 1 , 5 \, x^{3} + 1 \right\} \)is linearly independent.

Example 6 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 5 \, x + 1 , -4 \, x^{3} - 2 \, x + 3 , 6 \, x^{3} + 6 \, x^{2} - 12 \, x + 1 , 2 \, x^{3} - 6 \, x^{2} + 16 \, x - 7 \right\} \) is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 1 & -7 \\ 5 & -2 & -12 & 16 \\ -3 & 0 & 6 & -6 \\ -5 & -4 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & 1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 5 \, x + 1 , -4 \, x^{3} - 2 \, x + 3 , 6 \, x^{3} + 6 \, x^{2} - 12 \, x + 1 , 2 \, x^{3} - 6 \, x^{2} + 16 \, x - 7 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} - 3 \, x^{2} + 5 \, x + 1 \right) + y_{2} \left( -4 \, x^{3} - 2 \, x + 3 \right) + y_{3} \left( 6 \, x^{3} + 6 \, x^{2} - 12 \, x + 1 \right) + y_{4} \left( 2 \, x^{3} - 6 \, x^{2} + 16 \, x - 7 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 5 \, x + 1 , -4 \, x^{3} - 2 \, x + 3 , 6 \, x^{3} + 6 \, x^{2} - 12 \, x + 1 , 2 \, x^{3} - 6 \, x^{2} + 16 \, x - 7 \right\} \)is linearly dependent.

Example 7 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & -4 \\ 2 & -1 \end{array}\right] , \left[\begin{array}{cc} -1 & -2 \\ -5 & -5 \end{array}\right] , \left[\begin{array}{cc} 1 & -2 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 3 & 4 \\ -3 & 2 \end{array}\right] , \left[\begin{array}{cc} 3 & -4 \\ -3 & -3 \end{array}\right] , \left[\begin{array}{cc} -5 & 0 \\ 0 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -1 & -1 & 1 & 3 & 3 & -5 \\ -4 & -2 & -2 & 4 & -4 & 0 \\ 2 & -5 & 4 & -3 & -3 & 0 \\ -1 & -5 & 1 & 2 & -3 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{61}{45} & \frac{20}{9} \\ 0 & 1 & 0 & 0 & \frac{17}{15} & -\frac{5}{3} \\ 0 & 0 & 1 & 0 & \frac{49}{45} & -\frac{31}{9} \\ 0 & 0 & 0 & 1 & \frac{22}{15} & -\frac{1}{3} \end{array}\right] \]

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & -4 \\ 2 & -1 \end{array}\right] , \left[\begin{array}{cc} -1 & -2 \\ -5 & -5 \end{array}\right] , \left[\begin{array}{cc} 1 & -2 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 3 & 4 \\ -3 & 2 \end{array}\right] , \left[\begin{array}{cc} 3 & -4 \\ -3 & -3 \end{array}\right] , \left[\begin{array}{cc} -5 & 0 \\ 0 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -1 & -4 \\ 2 & -1 \end{array}\right] + y_{2} \left[\begin{array}{cc} -1 & -2 \\ -5 & -5 \end{array}\right] + y_{3} \left[\begin{array}{cc} 1 & -2 \\ 4 & 1 \end{array}\right] + y_{4} \left[\begin{array}{cc} 3 & 4 \\ -3 & 2 \end{array}\right] + y_{5} \left[\begin{array}{cc} 3 & -4 \\ -3 & -3 \end{array}\right] + y_{6} \left[\begin{array}{cc} -5 & 0 \\ 0 & 2 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & -4 \\ 2 & -1 \end{array}\right] , \left[\begin{array}{cc} -1 & -2 \\ -5 & -5 \end{array}\right] , \left[\begin{array}{cc} 1 & -2 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 3 & 4 \\ -3 & 2 \end{array}\right] , \left[\begin{array}{cc} 3 & -4 \\ -3 & -3 \end{array}\right] , \left[\begin{array}{cc} -5 & 0 \\ 0 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 8 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -x^{3} - 2 \, x^{2} - 6 \, x - 1 , 4 \, x^{3} - x^{2} + 3 \, x - 1 , -5 \, x^{3} - x^{2} - 9 \, x , -28 \, x^{3} - 14 \, x^{2} - 15 \, x + 14 \right\} \) is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 0 & 14 \\ -6 & 3 & -9 & -15 \\ -2 & -1 & -1 & -14 \\ -1 & 4 & -5 & -28 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ -x^{3} - 2 \, x^{2} - 6 \, x - 1 , 4 \, x^{3} - x^{2} + 3 \, x - 1 , -5 \, x^{3} - x^{2} - 9 \, x , -28 \, x^{3} - 14 \, x^{2} - 15 \, x + 14 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -x^{3} - 2 \, x^{2} - 6 \, x - 1 \right) + y_{2} \left( 4 \, x^{3} - x^{2} + 3 \, x - 1 \right) + y_{3} \left( -5 \, x^{3} - x^{2} - 9 \, x \right) + y_{4} \left( -28 \, x^{3} - 14 \, x^{2} - 15 \, x + 14 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -x^{3} - 2 \, x^{2} - 6 \, x - 1 , 4 \, x^{3} - x^{2} + 3 \, x - 1 , -5 \, x^{3} - x^{2} - 9 \, x , -28 \, x^{3} - 14 \, x^{2} - 15 \, x + 14 \right\} \)is linearly dependent.

Example 9 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 2 \, x^{3} - 3 \, x^{2} - 5 \, x , 4 \, x^{3} + 3 \, x^{2} + 4 \, x + 1 , 3 \, x^{3} - 5 \, x^{2} + 4 \, x + 2 , 3 \, x^{3} - 14 \, x^{2} - 10 \, x + 1 , x^{3} - 4 \, x^{2} - 4 \, x + 4 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} 0 & 1 & 2 & 1 & 4 \\ -5 & 4 & 4 & -10 & -4 \\ -3 & 3 & -5 & -14 & -4 \\ 2 & 4 & 3 & 3 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 2 \, x^{3} - 3 \, x^{2} - 5 \, x , 4 \, x^{3} + 3 \, x^{2} + 4 \, x + 1 , 3 \, x^{3} - 5 \, x^{2} + 4 \, x + 2 , 3 \, x^{3} - 14 \, x^{2} - 10 \, x + 1 , x^{3} - 4 \, x^{2} - 4 \, x + 4 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 2 \, x^{3} - 3 \, x^{2} - 5 \, x \right) + y_{2} \left( 4 \, x^{3} + 3 \, x^{2} + 4 \, x + 1 \right) + y_{3} \left( 3 \, x^{3} - 5 \, x^{2} + 4 \, x + 2 \right) + y_{4} \left( 3 \, x^{3} - 14 \, x^{2} - 10 \, x + 1 \right) + y_{5} \left( x^{3} - 4 \, x^{2} - 4 \, x + 4 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 2 \, x^{3} - 3 \, x^{2} - 5 \, x , 4 \, x^{3} + 3 \, x^{2} + 4 \, x + 1 , 3 \, x^{3} - 5 \, x^{2} + 4 \, x + 2 , 3 \, x^{3} - 14 \, x^{2} - 10 \, x + 1 , x^{3} - 4 \, x^{2} - 4 \, x + 4 \right\} \) spans \(\mathcal{P}_3\).

Example 10 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} + 2 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} - x^{2} - 2 \, x + 1 , -7 \, x^{3} + 3 \, x^{2} + 2 \, x - 1 , -14 \, x^{3} + 6 \, x^{2} + 4 \, x - 2 , 2 \, x^{3} + x^{2} + 3 \, x + 3 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} + 2 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} - x^{2} - 2 \, x + 1 , -7 \, x^{3} + 3 \, x^{2} + 2 \, x - 1 , -14 \, x^{3} + 6 \, x^{2} + 4 \, x - 2 , 2 \, x^{3} + x^{2} + 3 \, x + 3 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} + 2 \, x^{2} + 2 \, x - 1 \right) + y_{2} \left( 3 \, x^{3} - x^{2} - 2 \, x + 1 \right) + y_{3} \left( -7 \, x^{3} + 3 \, x^{2} + 2 \, x - 1 \right) + y_{4} \left( -14 \, x^{3} + 6 \, x^{2} + 4 \, x - 2 \right) + y_{5} \left( 2 \, x^{3} + x^{2} + 3 \, x + 3 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -5 \, x^{3} + 2 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} - x^{2} - 2 \, x + 1 , -7 \, x^{3} + 3 \, x^{2} + 2 \, x - 1 , -14 \, x^{3} + 6 \, x^{2} + 4 \, x - 2 , 2 \, x^{3} + x^{2} + 3 \, x + 3 \right\} \) does not span \(\mathcal{P}_3\).

Example 11 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 1 & -4 \\ -6 & 2 \end{array}\right] + y_{2} \left[\begin{array}{cc} -2 & -5 \\ -3 & -5 \end{array}\right] + y_{3} \left[\begin{array}{cc} 1 & -1 \\ -3 & -3 \end{array}\right] + y_{4} \left[\begin{array}{cc} 0 & -3 \\ 3 & -5 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 12 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -2 \\ -3 & -2 \end{array}\right] , \left[\begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & 1 \\ 0 & 0 \end{array}\right] , \left[\begin{array}{cc} 2 & 2 \\ 3 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -2 \\ -3 & -2 \end{array}\right] , \left[\begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & 1 \\ 0 & 0 \end{array}\right] , \left[\begin{array}{cc} 2 & 2 \\ 3 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 2 & -2 \\ -3 & -2 \end{array}\right] + y_{2} \left[\begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array}\right] + y_{3} \left[\begin{array}{cc} -3 & 1 \\ 0 & 0 \end{array}\right] + y_{4} \left[\begin{array}{cc} 2 & 2 \\ 3 & 2 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & -2 \\ -3 & -2 \end{array}\right] , \left[\begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array}\right] , \left[\begin{array}{cc} -3 & 1 \\ 0 & 0 \end{array}\right] , \left[\begin{array}{cc} 2 & 2 \\ 3 & 2 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 13 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 2 \, x , -6 \, x^{3} - 6 \, x^{2} - 3 \, x + 2 , -6 \, x^{3} + 3 \, x^{2} , -5 \, x^{3} + 2 \, x^{2} + 5 \, x - 3 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 2 \, x , -6 \, x^{3} - 6 \, x^{2} - 3 \, x + 2 , -6 \, x^{3} + 3 \, x^{2} , -5 \, x^{3} + 2 \, x^{2} + 5 \, x - 3 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} - 3 \, x^{2} + 2 \, x \right) + y_{2} \left( -6 \, x^{3} - 6 \, x^{2} - 3 \, x + 2 \right) + y_{3} \left( -6 \, x^{3} + 3 \, x^{2} \right) + y_{4} \left( -5 \, x^{3} + 2 \, x^{2} + 5 \, x - 3 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 2 \, x , -6 \, x^{3} - 6 \, x^{2} - 3 \, x + 2 , -6 \, x^{3} + 3 \, x^{2} , -5 \, x^{3} + 2 \, x^{2} + 5 \, x - 3 \right\} \)is linearly independent.

Example 14 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 2 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -9 & 0 \\ 9 & -6 \end{array}\right] , \left[\begin{array}{cc} 19 & -3 \\ -25 & 12 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & -3 & -9 & 19 \\ 3 & 0 & 0 & -3 \\ 4 & 3 & 9 & -25 \\ 2 & -2 & -6 & 12 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 3 & -7 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 2 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -9 & 0 \\ 9 & -6 \end{array}\right] , \left[\begin{array}{cc} 19 & -3 \\ -25 & 12 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 2 & 3 \\ 4 & 2 \end{array}\right] + y_{2} \left[\begin{array}{cc} -3 & 0 \\ 3 & -2 \end{array}\right] + y_{3} \left[\begin{array}{cc} -9 & 0 \\ 9 & -6 \end{array}\right] + y_{4} \left[\begin{array}{cc} 19 & -3 \\ -25 & 12 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 2 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -9 & 0 \\ 9 & -6 \end{array}\right] , \left[\begin{array}{cc} 19 & -3 \\ -25 & 12 \end{array}\right] \right\} \) does not span \(\mathrm{M}_{2,2}\).

Example 15 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 3 \, x^{3} + 4 \, x^{2} + 3 , -3 \, x^{3} + x^{2} + 2 \, x + 2 , x^{3} - 2 \, x^{2} + 4 \, x - 4 , -x^{3} - 3 \, x^{2} - 6 \, x - 1 , -3 \, x^{3} - 3 \, x^{2} + 2 \, x + 1 , -5 \, x^{3} + 3 \, x^{2} - 3 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 3 & 2 & -4 & -1 & 1 & -4 \\ 0 & 2 & 4 & -6 & 2 & -3 \\ 4 & 1 & -2 & -3 & -3 & 3 \\ 3 & -3 & 1 & -1 & -3 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & -\frac{93}{40} \\ 0 & 1 & 0 & -1 & 0 & \frac{647}{280} \\ 0 & 0 & 1 & -1 & 0 & -\frac{101}{280} \\ 0 & 0 & 0 & 0 & 1 & -\frac{173}{56} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 3 \, x^{3} + 4 \, x^{2} + 3 , -3 \, x^{3} + x^{2} + 2 \, x + 2 , x^{3} - 2 \, x^{2} + 4 \, x - 4 , -x^{3} - 3 \, x^{2} - 6 \, x - 1 , -3 \, x^{3} - 3 \, x^{2} + 2 \, x + 1 , -5 \, x^{3} + 3 \, x^{2} - 3 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 3 \, x^{3} + 4 \, x^{2} + 3 \right) + y_{2} \left( -3 \, x^{3} + x^{2} + 2 \, x + 2 \right) + y_{3} \left( x^{3} - 2 \, x^{2} + 4 \, x - 4 \right) + y_{4} \left( -x^{3} - 3 \, x^{2} - 6 \, x - 1 \right) + y_{5} \left( -3 \, x^{3} - 3 \, x^{2} + 2 \, x + 1 \right) + y_{6} \left( -5 \, x^{3} + 3 \, x^{2} - 3 \, x - 4 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 3 \, x^{3} + 4 \, x^{2} + 3 , -3 \, x^{3} + x^{2} + 2 \, x + 2 , x^{3} - 2 \, x^{2} + 4 \, x - 4 , -x^{3} - 3 \, x^{2} - 6 \, x - 1 , -3 \, x^{3} - 3 \, x^{2} + 2 \, x + 1 , -5 \, x^{3} + 3 \, x^{2} - 3 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

Example 16 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5 , 3 \, x^{3} + 3 \, x^{2} - 3 \, x + 4 , x^{3} - 3 \, x^{2} - 4 \, x - 4 , 4 \, x^{3} - 3 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} + 3 \, x^{2} + x + 2 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -5 & 4 & -4 & -1 & 2 \\ 3 & -3 & -4 & 2 & 1 \\ -3 & 3 & -3 & -3 & 3 \\ -5 & 3 & 1 & 4 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 46 \\ 0 & 1 & 0 & 0 & 59 \\ 0 & 0 & 1 & 0 & -\frac{8}{3} \\ 0 & 0 & 0 & 1 & \frac{44}{3} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5 , 3 \, x^{3} + 3 \, x^{2} - 3 \, x + 4 , x^{3} - 3 \, x^{2} - 4 \, x - 4 , 4 \, x^{3} - 3 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} + 3 \, x^{2} + x + 2 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5 \right) + y_{2} \left( 3 \, x^{3} + 3 \, x^{2} - 3 \, x + 4 \right) + y_{3} \left( x^{3} - 3 \, x^{2} - 4 \, x - 4 \right) + y_{4} \left( 4 \, x^{3} - 3 \, x^{2} + 2 \, x - 1 \right) + y_{5} \left( 3 \, x^{3} + 3 \, x^{2} + x + 2 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5 , 3 \, x^{3} + 3 \, x^{2} - 3 \, x + 4 , x^{3} - 3 \, x^{2} - 4 \, x - 4 , 4 \, x^{3} - 3 \, x^{2} + 2 \, x - 1 , 3 \, x^{3} + 3 \, x^{2} + x + 2 \right\} \) spans \(\mathcal{P}_3\).

Example 17 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{2} + 4 \, x - 3 , 2 \, x^{3} + 2 \, x^{2} - 4 \, x + 4 , -2 \, x^{3} - 5 \, x^{2} + x - 5 , x^{3} + 4 \, x^{2} - 5 , -4 \, x^{3} - x^{2} + 2 \, x - 5 , -5 \, x^{3} - 3 \, x^{2} + 2 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 4 & -5 & -5 & -5 & -4 \\ 4 & -4 & 1 & 0 & 2 & 2 \\ 1 & 2 & -5 & 4 & -1 & -3 \\ 0 & 2 & -2 & 1 & -4 & -5 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{111}{41} & -\frac{128}{41} \\ 0 & 1 & 0 & 0 & -\frac{142}{41} & -\frac{319}{82} \\ 0 & 0 & 1 & 0 & -\frac{42}{41} & -\frac{44}{41} \\ 0 & 0 & 0 & 1 & \frac{36}{41} & \frac{26}{41} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ x^{2} + 4 \, x - 3 , 2 \, x^{3} + 2 \, x^{2} - 4 \, x + 4 , -2 \, x^{3} - 5 \, x^{2} + x - 5 , x^{3} + 4 \, x^{2} - 5 , -4 \, x^{3} - x^{2} + 2 \, x - 5 , -5 \, x^{3} - 3 \, x^{2} + 2 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{2} + 4 \, x - 3 \right) + y_{2} \left( 2 \, x^{3} + 2 \, x^{2} - 4 \, x + 4 \right) + y_{3} \left( -2 \, x^{3} - 5 \, x^{2} + x - 5 \right) + y_{4} \left( x^{3} + 4 \, x^{2} - 5 \right) + y_{5} \left( -4 \, x^{3} - x^{2} + 2 \, x - 5 \right) + y_{6} \left( -5 \, x^{3} - 3 \, x^{2} + 2 \, x - 4 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ x^{2} + 4 \, x - 3 , 2 \, x^{3} + 2 \, x^{2} - 4 \, x + 4 , -2 \, x^{3} - 5 \, x^{2} + x - 5 , x^{3} + 4 \, x^{2} - 5 , -4 \, x^{3} - x^{2} + 2 \, x - 5 , -5 \, x^{3} - 3 \, x^{2} + 2 \, x - 4 \right\} \) spans \(\mathcal{P}_3\).

Example 18 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{2} - 5 \, x + 1 , x^{3} - x^{2} + x - 2 , -2 \, x^{3} + 3 \, x^{2} + x + 1 , -3 \, x^{3} + 4 \, x^{2} - 8 \, x + 7 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ x^{2} - 5 \, x + 1 , x^{3} - x^{2} + x - 2 , -2 \, x^{3} + 3 \, x^{2} + x + 1 , -3 \, x^{3} + 4 \, x^{2} - 8 \, x + 7 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{2} - 5 \, x + 1 \right) + y_{2} \left( x^{3} - x^{2} + x - 2 \right) + y_{3} \left( -2 \, x^{3} + 3 \, x^{2} + x + 1 \right) + y_{4} \left( -3 \, x^{3} + 4 \, x^{2} - 8 \, x + 7 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ x^{2} - 5 \, x + 1 , x^{3} - x^{2} + x - 2 , -2 \, x^{3} + 3 \, x^{2} + x + 1 , -3 \, x^{3} + 4 \, x^{2} - 8 \, x + 7 \right\} \) does not span \(\mathcal{P}_3\).

Example 19 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -2 & -2 \\ -6 & 5 \end{array}\right] + y_{2} \left[\begin{array}{cc} -6 & 0 \\ -5 & -2 \end{array}\right] + y_{3} \left[\begin{array}{cc} 5 & -6 \\ -3 & 3 \end{array}\right] + y_{4} \left[\begin{array}{cc} -9 & -8 \\ -19 & 4 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 20 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x - 2 , -3 \, x^{3} + x^{2} - 5 \, x - 5 , 3 \, x^{3} - x^{2} + 3 \, x + 4 , -3 \, x^{3} + x^{2} - 3 \, x - 4 , -4 \, x^{3} + 3 \, x^{2} + x + 1 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x - 2 , -3 \, x^{3} + x^{2} - 5 \, x - 5 , 3 \, x^{3} - x^{2} + 3 \, x + 4 , -3 \, x^{3} + x^{2} - 3 \, x - 4 , -4 \, x^{3} + 3 \, x^{2} + x + 1 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} + 4 \, x - 2 \right) + y_{2} \left( -3 \, x^{3} + x^{2} - 5 \, x - 5 \right) + y_{3} \left( 3 \, x^{3} - x^{2} + 3 \, x + 4 \right) + y_{4} \left( -3 \, x^{3} + x^{2} - 3 \, x - 4 \right) + y_{5} \left( -4 \, x^{3} + 3 \, x^{2} + x + 1 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x - 2 , -3 \, x^{3} + x^{2} - 5 \, x - 5 , 3 \, x^{3} - x^{2} + 3 \, x + 4 , -3 \, x^{3} + x^{2} - 3 \, x - 4 , -4 \, x^{3} + 3 \, x^{2} + x + 1 \right\} \) spans \(\mathcal{P}_3\).

Example 21 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -3 \, x^{3} + x^{2} - 5 \, x + 4 , 3 \, x^{3} + 4 \, x^{2} - 4 \, x , -4 \, x^{3} + 4 \, x^{2} + 2 \, x - 6 , 5 \, x^{3} - 4 \, x^{2} + 2 \, x - 3 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -3 \, x^{3} + x^{2} - 5 \, x + 4 , 3 \, x^{3} + 4 \, x^{2} - 4 \, x , -4 \, x^{3} + 4 \, x^{2} + 2 \, x - 6 , 5 \, x^{3} - 4 \, x^{2} + 2 \, x - 3 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -3 \, x^{3} + x^{2} - 5 \, x + 4 \right) + y_{2} \left( 3 \, x^{3} + 4 \, x^{2} - 4 \, x \right) + y_{3} \left( -4 \, x^{3} + 4 \, x^{2} + 2 \, x - 6 \right) + y_{4} \left( 5 \, x^{3} - 4 \, x^{2} + 2 \, x - 3 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -3 \, x^{3} + x^{2} - 5 \, x + 4 , 3 \, x^{3} + 4 \, x^{2} - 4 \, x , -4 \, x^{3} + 4 \, x^{2} + 2 \, x - 6 , 5 \, x^{3} - 4 \, x^{2} + 2 \, x - 3 \right\} \)is linearly independent.

Example 22 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -2 & 0 \\ -5 & 3 \end{array}\right] , \left[\begin{array}{cc} -3 & 5 \\ -2 & -1 \end{array}\right] , \left[\begin{array}{cc} 2 & 1 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 0 & -13 \\ -8 & -1 \end{array}\right] \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -2 & 0 \\ -5 & 3 \end{array}\right] , \left[\begin{array}{cc} -3 & 5 \\ -2 & -1 \end{array}\right] , \left[\begin{array}{cc} 2 & 1 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 0 & -13 \\ -8 & -1 \end{array}\right] \right\} \) is linearly independent.

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -2 & 0 \\ -5 & 3 \end{array}\right] + y_{2} \left[\begin{array}{cc} -3 & 5 \\ -2 & -1 \end{array}\right] + y_{3} \left[\begin{array}{cc} 2 & 1 \\ 4 & 1 \end{array}\right] + y_{4} \left[\begin{array}{cc} 0 & -13 \\ -8 & -1 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -2 & 0 \\ -5 & 3 \end{array}\right] , \left[\begin{array}{cc} -3 & 5 \\ -2 & -1 \end{array}\right] , \left[\begin{array}{cc} 2 & 1 \\ 4 & 1 \end{array}\right] , \left[\begin{array}{cc} 0 & -13 \\ -8 & -1 \end{array}\right] \right\} \)is linearly dependent.

Example 23 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 3 \, x - 3 , 2 \, x^{3} + x^{2} + x - 3 , -2 \, x^{3} - 5 \, x^{2} + 4 \, x - 5 , 6 \, x^{3} - 2 \, x^{2} + 11 \, x - 14 \right\} \) is linearly independent.

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -3 & -3 & -5 & -14 \\ 3 & 1 & 4 & 11 \\ 1 & 1 & -5 & -2 \\ 3 & 2 & -2 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 3 \, x - 3 , 2 \, x^{3} + x^{2} + x - 3 , -2 \, x^{3} - 5 \, x^{2} + 4 \, x - 5 , 6 \, x^{3} - 2 \, x^{2} + 11 \, x - 14 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 3 \, x^{3} + x^{2} + 3 \, x - 3 \right) + y_{2} \left( 2 \, x^{3} + x^{2} + x - 3 \right) + y_{3} \left( -2 \, x^{3} - 5 \, x^{2} + 4 \, x - 5 \right) + y_{4} \left( 6 \, x^{3} - 2 \, x^{2} + 11 \, x - 14 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 3 \, x - 3 , 2 \, x^{3} + x^{2} + x - 3 , -2 \, x^{3} - 5 \, x^{2} + 4 \, x - 5 , 6 \, x^{3} - 2 \, x^{2} + 11 \, x - 14 \right\} \)is linearly dependent.

Example 24 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 3 \\ 4 & -2 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ -4 & -5 \end{array}\right] , \left[\begin{array}{cc} -4 & 0 \\ 2 & 0 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ 3 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 3 \\ 4 & -2 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ -4 & -5 \end{array}\right] , \left[\begin{array}{cc} -4 & 0 \\ 2 & 0 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ 3 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -3 & 3 \\ 4 & -2 \end{array}\right] + y_{2} \left[\begin{array}{cc} -2 & 3 \\ -4 & -5 \end{array}\right] + y_{3} \left[\begin{array}{cc} -4 & 0 \\ 2 & 0 \end{array}\right] + y_{4} \left[\begin{array}{cc} -3 & -4 \\ 3 & 3 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -3 & 3 \\ 4 & -2 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ -4 & -5 \end{array}\right] , \left[\begin{array}{cc} -4 & 0 \\ 2 & 0 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ 3 & 3 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 25 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 2 \, x^{3} - 5 \, x^{2} - 5 \, x - 2 , 3 \, x^{3} - 5 \, x^{2} - 4 \, x + 3 , 3 \, x^{3} + 2 \, x^{2} + 1 , -4 \, x^{3} + 4 \, x^{2} - 2 \, x + 4 , -4 \, x^{3} - x^{2} + 2 , 4 \, x^{3} - 3 \, x^{2} - 5 \, x + 2 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -2 & 3 & 1 & 4 & 2 & 2 \\ -5 & -4 & 0 & -2 & 0 & -5 \\ -5 & -5 & 2 & 4 & -1 & -3 \\ 2 & 3 & 3 & -4 & -4 & 4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{80}{189} & \frac{32}{63} \\ 0 & 1 & 0 & 0 & \frac{25}{63} & \frac{11}{21} \\ 0 & 0 & 1 & 0 & -\frac{23}{21} & \frac{5}{7} \\ 0 & 0 & 0 & 1 & \frac{50}{189} & \frac{23}{126} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 2 \, x^{3} - 5 \, x^{2} - 5 \, x - 2 , 3 \, x^{3} - 5 \, x^{2} - 4 \, x + 3 , 3 \, x^{3} + 2 \, x^{2} + 1 , -4 \, x^{3} + 4 \, x^{2} - 2 \, x + 4 , -4 \, x^{3} - x^{2} + 2 , 4 \, x^{3} - 3 \, x^{2} - 5 \, x + 2 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 2 \, x^{3} - 5 \, x^{2} - 5 \, x - 2 \right) + y_{2} \left( 3 \, x^{3} - 5 \, x^{2} - 4 \, x + 3 \right) + y_{3} \left( 3 \, x^{3} + 2 \, x^{2} + 1 \right) + y_{4} \left( -4 \, x^{3} + 4 \, x^{2} - 2 \, x + 4 \right) + y_{5} \left( -4 \, x^{3} - x^{2} + 2 \right) + y_{6} \left( 4 \, x^{3} - 3 \, x^{2} - 5 \, x + 2 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 2 \, x^{3} - 5 \, x^{2} - 5 \, x - 2 , 3 \, x^{3} - 5 \, x^{2} - 4 \, x + 3 , 3 \, x^{3} + 2 \, x^{2} + 1 , -4 \, x^{3} + 4 \, x^{2} - 2 \, x + 4 , -4 \, x^{3} - x^{2} + 2 , 4 \, x^{3} - 3 \, x^{2} - 5 \, x + 2 \right\} \) spans \(\mathcal{P}_3\).

Example 26 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 5 & -3 \\ 5 & -4 \end{array}\right] + y_{2} \left[\begin{array}{cc} -3 & 4 \\ 2 & -3 \end{array}\right] + y_{3} \left[\begin{array}{cc} 3 & 1 \\ -2 & 2 \end{array}\right] + y_{4} \left[\begin{array}{cc} 0 & -3 \\ -2 & -4 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 27 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 4 \, x^{3} - x^{2} - 4 \, x + 3 , 4 \, x^{3} - 4 \, x^{2} + 2 \, x + 2 , -16 \, x^{3} + 7 \, x^{2} + 10 \, x - 11 , 4 \, x^{3} - 7 \, x^{2} + 8 \, x + 1 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 2 & -11 & 1 \\ -4 & 2 & 10 & 8 \\ -1 & -4 & 7 & -7 \\ 4 & 4 & -16 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & -1 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 4 \, x^{3} - x^{2} - 4 \, x + 3 , 4 \, x^{3} - 4 \, x^{2} + 2 \, x + 2 , -16 \, x^{3} + 7 \, x^{2} + 10 \, x - 11 , 4 \, x^{3} - 7 \, x^{2} + 8 \, x + 1 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 4 \, x^{3} - x^{2} - 4 \, x + 3 \right) + y_{2} \left( 4 \, x^{3} - 4 \, x^{2} + 2 \, x + 2 \right) + y_{3} \left( -16 \, x^{3} + 7 \, x^{2} + 10 \, x - 11 \right) + y_{4} \left( 4 \, x^{3} - 7 \, x^{2} + 8 \, x + 1 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 4 \, x^{3} - x^{2} - 4 \, x + 3 , 4 \, x^{3} - 4 \, x^{2} + 2 \, x + 2 , -16 \, x^{3} + 7 \, x^{2} + 10 \, x - 11 , 4 \, x^{3} - 7 \, x^{2} + 8 \, x + 1 \right\} \) does not span \(\mathcal{P}_3\).

Example 28 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{3} + 2 \, x^{2} - 2 \, x - 1 , -x^{3} + 5 \, x^{2} - 4 \, x , -3 \, x^{3} - 3 \, x^{2} - 5 \, x + 5 , 5 \, x^{3} - 3 \, x^{2} + 2 \, x - 5 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ x^{3} + 2 \, x^{2} - 2 \, x - 1 , -x^{3} + 5 \, x^{2} - 4 \, x , -3 \, x^{3} - 3 \, x^{2} - 5 \, x + 5 , 5 \, x^{3} - 3 \, x^{2} + 2 \, x - 5 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{3} + 2 \, x^{2} - 2 \, x - 1 \right) + y_{2} \left( -x^{3} + 5 \, x^{2} - 4 \, x \right) + y_{3} \left( -3 \, x^{3} - 3 \, x^{2} - 5 \, x + 5 \right) + y_{4} \left( 5 \, x^{3} - 3 \, x^{2} + 2 \, x - 5 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ x^{3} + 2 \, x^{2} - 2 \, x - 1 , -x^{3} + 5 \, x^{2} - 4 \, x , -3 \, x^{3} - 3 \, x^{2} - 5 \, x + 5 , 5 \, x^{3} - 3 \, x^{2} + 2 \, x - 5 \right\} \)is linearly independent.

Example 29 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 0 & 3 \\ -2 & -1 \end{array}\right] + y_{2} \left[\begin{array}{cc} -2 & -6 \\ 3 & 2 \end{array}\right] + y_{3} \left[\begin{array}{cc} 2 & -4 \\ 2 & -2 \end{array}\right] + y_{4} \left[\begin{array}{cc} 4 & -5 \\ -6 & 1 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 30 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} 4 & -1 \\ -6 & -5 \end{array}\right] , \left[\begin{array}{cc} -5 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} 3 & 4 \\ -4 & 0 \end{array}\right] , \left[\begin{array}{cc} 6 & -12 \\ 3 & 6 \end{array}\right] \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} 4 & -1 \\ -6 & -5 \end{array}\right] , \left[\begin{array}{cc} -5 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} 3 & 4 \\ -4 & 0 \end{array}\right] , \left[\begin{array}{cc} 6 & -12 \\ 3 & 6 \end{array}\right] \right\} \) is linearly independent.

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 4 & -1 \\ -6 & -5 \end{array}\right] + y_{2} \left[\begin{array}{cc} -5 & 0 \\ 3 & -2 \end{array}\right] + y_{3} \left[\begin{array}{cc} 3 & 4 \\ -4 & 0 \end{array}\right] + y_{4} \left[\begin{array}{cc} 6 & -12 \\ 3 & 6 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 31 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x^{2} - 3 , -5 \, x^{3} - 4 \, x^{2} - x + 3 , -5 \, x^{3} - 5 \, x^{2} - x - 4 , 15 \, x^{3} + 5 \, x^{2} + 2 \, x + 4 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x^{2} - 3 , -5 \, x^{3} - 4 \, x^{2} - x + 3 , -5 \, x^{3} - 5 \, x^{2} - x - 4 , 15 \, x^{3} + 5 \, x^{2} + 2 \, x + 4 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -5 \, x^{3} + 4 \, x^{2} - 3 \right) + y_{2} \left( -5 \, x^{3} - 4 \, x^{2} - x + 3 \right) + y_{3} \left( -5 \, x^{3} - 5 \, x^{2} - x - 4 \right) + y_{4} \left( 15 \, x^{3} + 5 \, x^{2} + 2 \, x + 4 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -5 \, x^{3} + 4 \, x^{2} - 3 , -5 \, x^{3} - 4 \, x^{2} - x + 3 , -5 \, x^{3} - 5 \, x^{2} - x - 4 , 15 \, x^{3} + 5 \, x^{2} + 2 \, x + 4 \right\} \) does not span \(\mathcal{P}_3\).

Example 32 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -2 \, x^{3} + 2 \, x^{2} - 4 \, x , -4 \, x^{2} - x - 5 , 4 \, x^{3} + 4 \, x^{2} + 10 \, x + 10 , 2 \, x^{3} + 11 \, x^{2} - x + 7 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -2 \, x^{3} + 2 \, x^{2} - 4 \, x , -4 \, x^{2} - x - 5 , 4 \, x^{3} + 4 \, x^{2} + 10 \, x + 10 , 2 \, x^{3} + 11 \, x^{2} - x + 7 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -2 \, x^{3} + 2 \, x^{2} - 4 \, x \right) + y_{2} \left( -4 \, x^{2} - x - 5 \right) + y_{3} \left( 4 \, x^{3} + 4 \, x^{2} + 10 \, x + 10 \right) + y_{4} \left( 2 \, x^{3} + 11 \, x^{2} - x + 7 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -2 \, x^{3} + 2 \, x^{2} - 4 \, x , -4 \, x^{2} - x - 5 , 4 \, x^{3} + 4 \, x^{2} + 10 \, x + 10 , 2 \, x^{3} + 11 \, x^{2} - x + 7 \right\} \)is linearly dependent.

Example 33 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -6 \, x^{3} - 5 \, x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 6 \, x + 2 , -2 \, x^{3} - 4 \, x^{2} - 2 \, x , 2 \, x^{3} + 8 \, x^{2} + 7 \, x + 2 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -6 \, x^{3} - 5 \, x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 6 \, x + 2 , -2 \, x^{3} - 4 \, x^{2} - 2 \, x , 2 \, x^{3} + 8 \, x^{2} + 7 \, x + 2 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -6 \, x^{3} - 5 \, x^{2} - x - 4 \right) + y_{2} \left( 4 \, x^{3} - 3 \, x^{2} - 6 \, x + 2 \right) + y_{3} \left( -2 \, x^{3} - 4 \, x^{2} - 2 \, x \right) + y_{4} \left( 2 \, x^{3} + 8 \, x^{2} + 7 \, x + 2 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -6 \, x^{3} - 5 \, x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 6 \, x + 2 , -2 \, x^{3} - 4 \, x^{2} - 2 \, x , 2 \, x^{3} + 8 \, x^{2} + 7 \, x + 2 \right\} \)is linearly dependent.

Example 34 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 3 \, x^{3} - 3 \, x^{2} - 3 \, x + 1 , 4 \, x^{3} - 4 \, x^{2} + x - 2 , -3 \, x^{3} - 5 \, x^{2} - 4 \, x - 2 , -3 \, x^{3} + 27 \, x^{2} + 9 \, x + 12 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -2 & 12 \\ -3 & 1 & -4 & 9 \\ -3 & -4 & -5 & 27 \\ 3 & 4 & -3 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 3 \, x^{3} - 3 \, x^{2} - 3 \, x + 1 , 4 \, x^{3} - 4 \, x^{2} + x - 2 , -3 \, x^{3} - 5 \, x^{2} - 4 \, x - 2 , -3 \, x^{3} + 27 \, x^{2} + 9 \, x + 12 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 3 \, x^{3} - 3 \, x^{2} - 3 \, x + 1 \right) + y_{2} \left( 4 \, x^{3} - 4 \, x^{2} + x - 2 \right) + y_{3} \left( -3 \, x^{3} - 5 \, x^{2} - 4 \, x - 2 \right) + y_{4} \left( -3 \, x^{3} + 27 \, x^{2} + 9 \, x + 12 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 3 \, x^{3} - 3 \, x^{2} - 3 \, x + 1 , 4 \, x^{3} - 4 \, x^{2} + x - 2 , -3 \, x^{3} - 5 \, x^{2} - 4 \, x - 2 , -3 \, x^{3} + 27 \, x^{2} + 9 \, x + 12 \right\} \) does not span \(\mathcal{P}_3\).

Example 35 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -x^{3} - 5 \, x^{2} - 3 \, x + 4 , x^{3} + 4 \, x^{2} + x - 5 , 2 \, x^{3} + 4 \, x + 2 , 2 \, x^{3} - 10 \, x^{2} + 2 \, x + 12 , -5 \, x^{2} - 5 \, x , -4 \, x^{2} + x \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} 4 & -5 & 2 & 12 & 0 & 0 \\ -3 & 1 & 4 & 2 & -5 & 1 \\ -5 & 4 & 0 & -10 & -5 & -4 \\ -1 & 1 & 2 & 2 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 2 & 0 & -30 \\ 0 & 1 & 0 & 0 & 0 & -25 \\ 0 & 0 & 1 & 2 & 0 & -\frac{5}{2} \\ 0 & 0 & 0 & 0 & 1 & \frac{54}{5} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ -x^{3} - 5 \, x^{2} - 3 \, x + 4 , x^{3} + 4 \, x^{2} + x - 5 , 2 \, x^{3} + 4 \, x + 2 , 2 \, x^{3} - 10 \, x^{2} + 2 \, x + 12 , -5 \, x^{2} - 5 \, x , -4 \, x^{2} + x \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -x^{3} - 5 \, x^{2} - 3 \, x + 4 \right) + y_{2} \left( x^{3} + 4 \, x^{2} + x - 5 \right) + y_{3} \left( 2 \, x^{3} + 4 \, x + 2 \right) + y_{4} \left( 2 \, x^{3} - 10 \, x^{2} + 2 \, x + 12 \right) + y_{5} \left( -5 \, x^{2} - 5 \, x \right) + y_{6} \left( -4 \, x^{2} + x \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -x^{3} - 5 \, x^{2} - 3 \, x + 4 , x^{3} + 4 \, x^{2} + x - 5 , 2 \, x^{3} + 4 \, x + 2 , 2 \, x^{3} - 10 \, x^{2} + 2 \, x + 12 , -5 \, x^{2} - 5 \, x , -4 \, x^{2} + x \right\} \) spans \(\mathcal{P}_3\).

Example 36 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -3 \, x^{3} + 4 \, x^{2} - 5 \, x + 4 , -x^{3} + 2 \, x^{2} - 6 \, x - 2 , -4 \, x^{3} - x^{2} - 4 \, x + 2 , -2 \, x^{3} + 4 \, x^{2} - 2 \, x - 3 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -3 \, x^{3} + 4 \, x^{2} - 5 \, x + 4 , -x^{3} + 2 \, x^{2} - 6 \, x - 2 , -4 \, x^{3} - x^{2} - 4 \, x + 2 , -2 \, x^{3} + 4 \, x^{2} - 2 \, x - 3 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -3 \, x^{3} + 4 \, x^{2} - 5 \, x + 4 \right) + y_{2} \left( -x^{3} + 2 \, x^{2} - 6 \, x - 2 \right) + y_{3} \left( -4 \, x^{3} - x^{2} - 4 \, x + 2 \right) + y_{4} \left( -2 \, x^{3} + 4 \, x^{2} - 2 \, x - 3 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -3 \, x^{3} + 4 \, x^{2} - 5 \, x + 4 , -x^{3} + 2 \, x^{2} - 6 \, x - 2 , -4 \, x^{3} - x^{2} - 4 \, x + 2 , -2 \, x^{3} + 4 \, x^{2} - 2 \, x - 3 \right\} \)is linearly independent.

Example 37 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{3} - 5 \, x + 4 , -5 \, x^{3} - 6 \, x^{2} + 5 \, x , -6 \, x^{3} - 4 \, x^{2} + 3 \, x - 3 , 3 \, x^{3} + 3 \, x^{2} + 4 \, x + 4 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ x^{3} - 5 \, x + 4 , -5 \, x^{3} - 6 \, x^{2} + 5 \, x , -6 \, x^{3} - 4 \, x^{2} + 3 \, x - 3 , 3 \, x^{3} + 3 \, x^{2} + 4 \, x + 4 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{3} - 5 \, x + 4 \right) + y_{2} \left( -5 \, x^{3} - 6 \, x^{2} + 5 \, x \right) + y_{3} \left( -6 \, x^{3} - 4 \, x^{2} + 3 \, x - 3 \right) + y_{4} \left( 3 \, x^{3} + 3 \, x^{2} + 4 \, x + 4 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ x^{3} - 5 \, x + 4 , -5 \, x^{3} - 6 \, x^{2} + 5 \, x , -6 \, x^{3} - 4 \, x^{2} + 3 \, x - 3 , 3 \, x^{3} + 3 \, x^{2} + 4 \, x + 4 \right\} \)is linearly independent.

Example 38 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{3} - 3 \, x^{2} - 4 \, x - 2 , 2 \, x^{3} - x^{2} - 3 \, x - 1 , -7 \, x^{3} + 6 \, x^{2} + 13 \, x + 5 , 10 \, x^{3} - 5 \, x^{2} - 15 \, x - 5 , -x^{3} + 3 \, x^{2} - 5 \, x - 2 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -2 & -1 & 5 & -5 & -2 \\ -4 & -3 & 13 & -15 & -5 \\ -3 & -1 & 6 & -5 & 3 \\ 1 & 2 & -7 & 10 & -1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -3 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ x^{3} - 3 \, x^{2} - 4 \, x - 2 , 2 \, x^{3} - x^{2} - 3 \, x - 1 , -7 \, x^{3} + 6 \, x^{2} + 13 \, x + 5 , 10 \, x^{3} - 5 \, x^{2} - 15 \, x - 5 , -x^{3} + 3 \, x^{2} - 5 \, x - 2 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{3} - 3 \, x^{2} - 4 \, x - 2 \right) + y_{2} \left( 2 \, x^{3} - x^{2} - 3 \, x - 1 \right) + y_{3} \left( -7 \, x^{3} + 6 \, x^{2} + 13 \, x + 5 \right) + y_{4} \left( 10 \, x^{3} - 5 \, x^{2} - 15 \, x - 5 \right) + y_{5} \left( -x^{3} + 3 \, x^{2} - 5 \, x - 2 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ x^{3} - 3 \, x^{2} - 4 \, x - 2 , 2 \, x^{3} - x^{2} - 3 \, x - 1 , -7 \, x^{3} + 6 \, x^{2} + 13 \, x + 5 , 10 \, x^{3} - 5 \, x^{2} - 15 \, x - 5 , -x^{3} + 3 \, x^{2} - 5 \, x - 2 \right\} \) does not span \(\mathcal{P}_3\).

Example 39 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} 0 & 2 \\ -1 & -5 \end{array}\right] , \left[\begin{array}{cc} -6 & 2 \\ -1 & 4 \end{array}\right] , \left[\begin{array}{cc} 0 & 1 \\ 5 & -1 \end{array}\right] , \left[\begin{array}{cc} 0 & 6 \\ 8 & -12 \end{array}\right] \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} 0 & 2 \\ -1 & -5 \end{array}\right] , \left[\begin{array}{cc} -6 & 2 \\ -1 & 4 \end{array}\right] , \left[\begin{array}{cc} 0 & 1 \\ 5 & -1 \end{array}\right] , \left[\begin{array}{cc} 0 & 6 \\ 8 & -12 \end{array}\right] \right\} \) is linearly independent.

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 0 & 2 \\ -1 & -5 \end{array}\right] + y_{2} \left[\begin{array}{cc} -6 & 2 \\ -1 & 4 \end{array}\right] + y_{3} \left[\begin{array}{cc} 0 & 1 \\ 5 & -1 \end{array}\right] + y_{4} \left[\begin{array}{cc} 0 & 6 \\ 8 & -12 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 40 πŸ”—

Consider the statement

The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & 2 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -1 & -4 \\ 0 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 6 \\ -3 & -6 \end{array}\right] , \left[\begin{array}{cc} 11 & 20 \\ -12 & -20 \end{array}\right] , \left[\begin{array}{cc} -1 & -3 \\ 4 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 2 & 1 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -1 & -1 & 3 & 11 & -1 & 0 \\ 2 & -4 & 6 & 20 & -3 & -1 \\ 3 & 0 & -3 & -12 & 4 & 2 \\ -2 & 4 & -6 & -20 & -5 & 1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -1 & -4 & 0 & 0 \\ 0 & 1 & -2 & -7 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

  1. The statement

    The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & 2 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -1 & -4 \\ 0 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 6 \\ -3 & -6 \end{array}\right] , \left[\begin{array}{cc} 11 & 20 \\ -12 & -20 \end{array}\right] , \left[\begin{array}{cc} -1 & -3 \\ 4 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 2 & 1 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -1 & 2 \\ 3 & -2 \end{array}\right] + y_{2} \left[\begin{array}{cc} -1 & -4 \\ 0 & 4 \end{array}\right] + y_{3} \left[\begin{array}{cc} 3 & 6 \\ -3 & -6 \end{array}\right] + y_{4} \left[\begin{array}{cc} 11 & 20 \\ -12 & -20 \end{array}\right] + y_{5} \left[\begin{array}{cc} -1 & -3 \\ 4 & -5 \end{array}\right] + y_{6} \left[\begin{array}{cc} 0 & -1 \\ 2 & 1 \end{array}\right] =B\]

    has a solution for every \(B \in \mathrm{M}_{2,2}\).

  2. The set of matrices \( \left\{ \left[\begin{array}{cc} -1 & 2 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -1 & -4 \\ 0 & 4 \end{array}\right] , \left[\begin{array}{cc} 3 & 6 \\ -3 & -6 \end{array}\right] , \left[\begin{array}{cc} 11 & 20 \\ -12 & -20 \end{array}\right] , \left[\begin{array}{cc} -1 & -3 \\ 4 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & -1 \\ 2 & 1 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\).

Example 41 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -2 \, x^{3} + 3 \, x^{2} - 4 \, x - 2 , -x^{3} - 2 \, x^{2} + x - 1 , -4 \, x^{3} - 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 3 \, x^{2} + 3 \, x - 4 , -5 \, x^{3} - 2 \, x^{2} - 3 \, x - 5 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc} -2 & -1 & 4 & -4 & -5 \\ -4 & 1 & 2 & 3 & -3 \\ 3 & -2 & -5 & -3 & -2 \\ -2 & -1 & -4 & -4 & -5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{32}{23} \\ 0 & 1 & 0 & 0 & \frac{83}{23} \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & -\frac{8}{23} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ -2 \, x^{3} + 3 \, x^{2} - 4 \, x - 2 , -x^{3} - 2 \, x^{2} + x - 1 , -4 \, x^{3} - 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 3 \, x^{2} + 3 \, x - 4 , -5 \, x^{3} - 2 \, x^{2} - 3 \, x - 5 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -2 \, x^{3} + 3 \, x^{2} - 4 \, x - 2 \right) + y_{2} \left( -x^{3} - 2 \, x^{2} + x - 1 \right) + y_{3} \left( -4 \, x^{3} - 5 \, x^{2} + 2 \, x + 4 \right) + y_{4} \left( -4 \, x^{3} - 3 \, x^{2} + 3 \, x - 4 \right) + y_{5} \left( -5 \, x^{3} - 2 \, x^{2} - 3 \, x - 5 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ -2 \, x^{3} + 3 \, x^{2} - 4 \, x - 2 , -x^{3} - 2 \, x^{2} + x - 1 , -4 \, x^{3} - 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 3 \, x^{2} + 3 \, x - 4 , -5 \, x^{3} - 2 \, x^{2} - 3 \, x - 5 \right\} \) spans \(\mathcal{P}_3\).

Example 42 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 3 \, x^{3} - x^{2} - x + 2 , -2 \, x^{3} + 2 \, x^{2} + 3 \, x - 5 , -6 \, x^{3} + 2 \, x^{2} - 2 \, x + 5 , 2 \, x^{3} - 5 \, x^{2} + 5 \, x - 4 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ 3 \, x^{3} - x^{2} - x + 2 , -2 \, x^{3} + 2 \, x^{2} + 3 \, x - 5 , -6 \, x^{3} + 2 \, x^{2} - 2 \, x + 5 , 2 \, x^{3} - 5 \, x^{2} + 5 \, x - 4 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 3 \, x^{3} - x^{2} - x + 2 \right) + y_{2} \left( -2 \, x^{3} + 2 \, x^{2} + 3 \, x - 5 \right) + y_{3} \left( -6 \, x^{3} + 2 \, x^{2} - 2 \, x + 5 \right) + y_{4} \left( 2 \, x^{3} - 5 \, x^{2} + 5 \, x - 4 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ 3 \, x^{3} - x^{2} - x + 2 , -2 \, x^{3} + 2 \, x^{2} + 3 \, x - 5 , -6 \, x^{3} + 2 \, x^{2} - 2 \, x + 5 , 2 \, x^{3} - 5 \, x^{2} + 5 \, x - 4 \right\} \)is linearly independent.

Example 43 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} -2 & 3 \\ -2 & -6 \end{array}\right] + y_{2} \left[\begin{array}{cc} 2 & -4 \\ 3 & 0 \end{array}\right] + y_{3} \left[\begin{array}{cc} -5 & -3 \\ -3 & -3 \end{array}\right] + y_{4} \left[\begin{array}{cc} -2 & -1 \\ 0 & 5 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 44 πŸ”—

Consider the statement

The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 4 \, x - 5 , -3 \, x^{3} - x^{2} + 3 \, x + 2 , 4 \, x^{3} - x^{2} - 2 \, x + 3 , -3 \, x^{3} + x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 2 \, x , 2 \, x^{3} + 4 \, x^{2} - 2 \, x - 2 \right\} \) spans \(\mathcal{P}_3\).

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

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccccc} -5 & 2 & 3 & -4 & 0 & -2 \\ 4 & 3 & -2 & -1 & -2 & -2 \\ 1 & -1 & -1 & 1 & -3 & 4 \\ 3 & -3 & 4 & -3 & 4 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{14}{13} & -\frac{12}{13} \\ 0 & 1 & 0 & 0 & \frac{291}{104} & -\frac{53}{13} \\ 0 & 0 & 1 & 0 & \frac{277}{52} & -\frac{64}{13} \\ 0 & 0 & 0 & 1 & \frac{421}{104} & -\frac{53}{13} \end{array}\right] \]

  1. The statement

    The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 4 \, x - 5 , -3 \, x^{3} - x^{2} + 3 \, x + 2 , 4 \, x^{3} - x^{2} - 2 \, x + 3 , -3 \, x^{3} + x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 2 \, x , 2 \, x^{3} + 4 \, x^{2} - 2 \, x - 2 \right\} \) spans \(\mathcal{P}_3\).

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( 3 \, x^{3} + x^{2} + 4 \, x - 5 \right) + y_{2} \left( -3 \, x^{3} - x^{2} + 3 \, x + 2 \right) + y_{3} \left( 4 \, x^{3} - x^{2} - 2 \, x + 3 \right) + y_{4} \left( -3 \, x^{3} + x^{2} - x - 4 \right) + y_{5} \left( 4 \, x^{3} - 3 \, x^{2} - 2 \, x \right) + y_{6} \left( 2 \, x^{3} + 4 \, x^{2} - 2 \, x - 2 \right) =f\]

    has a solution for every \(f \in \mathcal{P}_3\).

  2. The set of polynomials \( \left\{ 3 \, x^{3} + x^{2} + 4 \, x - 5 , -3 \, x^{3} - x^{2} + 3 \, x + 2 , 4 \, x^{3} - x^{2} - 2 \, x + 3 , -3 \, x^{3} + x^{2} - x - 4 , 4 \, x^{3} - 3 \, x^{2} - 2 \, x , 2 \, x^{3} + 4 \, x^{2} - 2 \, x - 2 \right\} \) spans \(\mathcal{P}_3\).

Example 45 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -3 \, x^{3} + 3 , -3 \, x^{3} - 3 \, x^{2} + 5 \, x - 6 , x^{3} - 3 \, x , 6 \, x^{3} + 6 \, x^{2} - 10 \, x + 12 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -3 \, x^{3} + 3 , -3 \, x^{3} - 3 \, x^{2} + 5 \, x - 6 , x^{3} - 3 \, x , 6 \, x^{3} + 6 \, x^{2} - 10 \, x + 12 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -3 \, x^{3} + 3 \right) + y_{2} \left( -3 \, x^{3} - 3 \, x^{2} + 5 \, x - 6 \right) + y_{3} \left( x^{3} - 3 \, x \right) + y_{4} \left( 6 \, x^{3} + 6 \, x^{2} - 10 \, x + 12 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -3 \, x^{3} + 3 , -3 \, x^{3} - 3 \, x^{2} + 5 \, x - 6 , x^{3} - 3 \, x , 6 \, x^{3} + 6 \, x^{2} - 10 \, x + 12 \right\} \)is linearly dependent.

Example 46 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -3 \, x^{3} + 5 \, x + 1 , 2 \, x^{3} + 3 \, x + 1 , -6 \, x^{2} - 4 \, x - 6 , -7 \, x^{3} + 6 \, x^{2} + 3 \, x + 5 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -3 \, x^{3} + 5 \, x + 1 , 2 \, x^{3} + 3 \, x + 1 , -6 \, x^{2} - 4 \, x - 6 , -7 \, x^{3} + 6 \, x^{2} + 3 \, x + 5 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -3 \, x^{3} + 5 \, x + 1 \right) + y_{2} \left( 2 \, x^{3} + 3 \, x + 1 \right) + y_{3} \left( -6 \, x^{2} - 4 \, x - 6 \right) + y_{4} \left( -7 \, x^{3} + 6 \, x^{2} + 3 \, x + 5 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -3 \, x^{3} + 5 \, x + 1 , 2 \, x^{3} + 3 \, x + 1 , -6 \, x^{2} - 4 \, x - 6 , -7 \, x^{3} + 6 \, x^{2} + 3 \, x + 5 \right\} \)is linearly dependent.

Example 47 πŸ”—

Consider the statement

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

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

Answer:

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

  1. The statement

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

    is equivalent to the statement

    The matrix equation

    \[ y_{1} \left[\begin{array}{cc} 5 & 2 \\ -4 & -2 \end{array}\right] + y_{2} \left[\begin{array}{cc} 0 & -5 \\ 3 & -5 \end{array}\right] + y_{3} \left[\begin{array}{cc} -4 & 4 \\ 4 & 1 \end{array}\right] + y_{4} \left[\begin{array}{cc} 4 & 1 \\ 3 & 4 \end{array}\right] = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \]

    has no nontrivial solutions.

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

Example 48 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -4 \, x^{3} - 5 \, x^{2} - 4 \, x - 3 , -4 \, x^{3} - x^{2} - 2 \, x - 4 , 5 \, x^{3} - 4 , -6 \, x^{3} - 5 \, x^{2} - 6 \, x + 2 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -4 \, x^{3} - 5 \, x^{2} - 4 \, x - 3 , -4 \, x^{3} - x^{2} - 2 \, x - 4 , 5 \, x^{3} - 4 , -6 \, x^{3} - 5 \, x^{2} - 6 \, x + 2 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -4 \, x^{3} - 5 \, x^{2} - 4 \, x - 3 \right) + y_{2} \left( -4 \, x^{3} - x^{2} - 2 \, x - 4 \right) + y_{3} \left( 5 \, x^{3} - 4 \right) + y_{4} \left( -6 \, x^{3} - 5 \, x^{2} - 6 \, x + 2 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -4 \, x^{3} - 5 \, x^{2} - 4 \, x - 3 , -4 \, x^{3} - x^{2} - 2 \, x - 4 , 5 \, x^{3} - 4 , -6 \, x^{3} - 5 \, x^{2} - 6 \, x + 2 \right\} \)is linearly independent.

Example 49 πŸ”—

Consider the statement

The set of polynomials \( \left\{ -4 \, x^{3} + x^{2} + 3 \, x - 1 , 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 6 \, x^{2} + 4 , -14 \, x^{2} - 6 \, x + 10 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ -4 \, x^{3} + x^{2} + 3 \, x - 1 , 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 6 \, x^{2} + 4 , -14 \, x^{2} - 6 \, x + 10 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( -4 \, x^{3} + x^{2} + 3 \, x - 1 \right) + y_{2} \left( 5 \, x^{2} + 2 \, x + 4 \right) + y_{3} \left( -4 \, x^{3} - 6 \, x^{2} + 4 \right) + y_{4} \left( -14 \, x^{2} - 6 \, x + 10 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ -4 \, x^{3} + x^{2} + 3 \, x - 1 , 5 \, x^{2} + 2 \, x + 4 , -4 \, x^{3} - 6 \, x^{2} + 4 , -14 \, x^{2} - 6 \, x + 10 \right\} \)is linearly dependent.

Example 50 πŸ”—

Consider the statement

The set of polynomials \( \left\{ x^{3} + x^{2} - 5 \, x - 1 , -4 \, x^{3} - 2 \, x^{2} + x + 2 , 4 \, x^{3} - x^{2} - 3 \, x - 5 , 2 \, x^{3} - 5 \, x^{2} - 3 \, x - 1 \right\} \) is linearly independent.

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

Answer:

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

  1. The statement

    The set of polynomials \( \left\{ x^{3} + x^{2} - 5 \, x - 1 , -4 \, x^{3} - 2 \, x^{2} + x + 2 , 4 \, x^{3} - x^{2} - 3 \, x - 5 , 2 \, x^{3} - 5 \, x^{2} - 3 \, x - 1 \right\} \) is linearly independent.

    is equivalent to the statement

    The polynomial equation

    \[ y_{1} \left( x^{3} + x^{2} - 5 \, x - 1 \right) + y_{2} \left( -4 \, x^{3} - 2 \, x^{2} + x + 2 \right) + y_{3} \left( 4 \, x^{3} - x^{2} - 3 \, x - 5 \right) + y_{4} \left( 2 \, x^{3} - 5 \, x^{2} - 3 \, x - 1 \right) = 0 \]

    has no nontrivial solutions.

  2. The set of polynomials \( \left\{ x^{3} + x^{2} - 5 \, x - 1 , -4 \, x^{3} - 2 \, x^{2} + x + 2 , 4 \, x^{3} - x^{2} - 3 \, x - 5 , 2 \, x^{3} - 5 \, x^{2} - 3 \, x - 1 \right\} \)is linearly independent.