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Section 2.8 Polynomial and Matrix Spaces (V8)

Observation 2.8.2.

We've already been taking advantage of the previous fact by converting polynomials and matrices into Euclidean vectors. Since \(\P^3\) and \(M_{2,2}\) are both four-dimensional:

\begin{equation*} 4x^3+0x^2-1x+5 \leftrightarrow \left[\begin{array}{c} 4\\0\\-1\\5 \end{array}\right] \leftrightarrow \left[\begin{array}{cc} 4&0\\-1&5 \end{array}\right] \end{equation*}
Activity 2.8.1.

Suppose \(W\) is a subspace of \(\P^8\text{,}\) and you know that the set \(\{ x^3+x, x^2+1, x^4-x \}\) is a linearly independent subset of \(W\text{.}\) What can you conclude about \(W\text{?}\)

  1. The dimension of \(W\) is 3 or less.

  2. The dimension of \(W\) is exactly 3.

  3. The dimension of \(W\) is 3 or more.

Activity 2.8.2.

Suppose \(W\) is a subspace of \(\P^8\text{,}\) and you know that \(W\) is spanned by the six vectors

\begin{equation*} \{ x^4-x,x^3+x,x^3+x+1,x^4+2x,x^3,2x+1\}. \end{equation*}

What can you conclude about \(W\text{?}\)

  1. The dimension of \(W\) is 6 or less.

  2. The dimension of \(W\) is exactly 3.

  3. The dimension of \(W\) is 6 or more.

Observation 2.8.3.

The space of polynomials \(\P\) (of any degree) has the basis \(\{1,x,x^2,x^3,\dots\}\text{,}\) so it is a natural example of an infinite-dimensional vector space.

Since \(\P\) and other infinite-dimensional spaces cannot be treated as an isomorphic finite-dimensional Euclidean space \(\IR^n\text{,}\) vectors in such spaces cannot be studied by converting them into Euclidean vectors. Fortunately, most of the examples we will be interested in for this course will be finite-dimensional.

Exercises 2.8.1 Exercises

1.

Consider the statement

  1. Write an equivalent statement using a polynomial equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} -6 & 2 & -4 & -4 \\ -1 & 5 & 1 & -10 \\ 2 & 3 & 3 & -6 \\ -4 & -5 & -5 & 10 \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] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of polynomials \(\left\{ -4 \, x^{3} + 2 \, x^{2} - x - 6 , -5 \, x^{3} + 3 \, x^{2} + 5 \, x + 2 , -5 \, x^{3} + 3 \, x^{2} + x - 4 , 10 \, x^{3} - 6 \, x^{2} - 10 \, x - 4 \right\} \)is linearly dependent.
2.

Consider the statement

  1. Write an equivalent statement using a matrix equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 0 & 5 & -4 & -3 \\ 1 & 3 & 3 & -4 \\ -6 & -5 & -5 & 1 \\ 0 & 5 & -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] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of matrices \(\left\{ \left[\begin{array}{cc} 0 & 1 \\ -6 & 0 \end{array}\right] , \left[\begin{array}{cc} 5 & 3 \\ -5 & 5 \end{array}\right] , \left[\begin{array}{cc} -4 & 3 \\ -5 & -2 \end{array}\right] , \left[\begin{array}{cc} -3 & -4 \\ 1 & -4 \end{array}\right] \right\} \)is linearly independent.
3.

Consider the statement

  1. Write an equivalent statement using a matrix equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} -4 & -4 & 4 & 17 \\ 0 & 4 & 0 & -7 \\ 3 & -2 & -3 & 0 \\ -2 & -1 & 2 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of matrices \(\left\{ \left[\begin{array}{cc} -4 & 0 \\ 3 & -2 \end{array}\right] , \left[\begin{array}{cc} -4 & 4 \\ -2 & -1 \end{array}\right] , \left[\begin{array}{cc} 4 & 0 \\ -3 & 2 \end{array}\right] , \left[\begin{array}{cc} 17 & -7 \\ 0 & 8 \end{array}\right] \right\} \)is linearly dependent.
4.

Consider the statement

  1. Write an equivalent statement using a matrix equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccccc} 2 & -4 & -4 & -3 & 4 & -2 \\ 0 & 4 & -4 & 0 & 0 & 3 \\ -4 & -1 & -1 & 2 & -3 & 2 \\ -2 & -3 & -3 & -5 & -2 & -1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{9}{8} & -\frac{3}{8} \\ 0 & 1 & 0 & 0 & -\frac{4}{11} & \frac{37}{88} \\ 0 & 0 & 1 & 0 & -\frac{4}{11} & -\frac{29}{88} \\ 0 & 0 & 0 & 1 & \frac{17}{44} & \frac{13}{44} \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of matrices \(\left\{ \left[\begin{array}{cc} 2 & 0 \\ -4 & -2 \end{array}\right] , \left[\begin{array}{cc} -4 & 4 \\ -1 & -3 \end{array}\right] , \left[\begin{array}{cc} -4 & -4 \\ -1 & -3 \end{array}\right] , \left[\begin{array}{cc} -3 & 0 \\ 2 & -5 \end{array}\right] , \left[\begin{array}{cc} 4 & 0 \\ -3 & -2 \end{array}\right] , \left[\begin{array}{cc} -2 & 3 \\ 2 & -1 \end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\text{.}\)
5.

Consider the statement

  1. Write an equivalent statement using a matrix equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 3 & 0 & -5 & 6 \\ 4 & 3 & -2 & 8 \\ -3 & -2 & -5 & -6 \\ -5 & -2 & 1 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of matrices \(\left\{ \left[\begin{array}{cc} 3 & 4 \\ -3 & -5 \end{array}\right] , \left[\begin{array}{cc} 0 & 3 \\ -2 & -2 \end{array}\right] , \left[\begin{array}{cc} -5 & -2 \\ -5 & 1 \end{array}\right] , \left[\begin{array}{cc} 6 & 8 \\ -6 & -10 \end{array}\right] \right\} \) does not span \(\mathrm{M}_{2,2}\text{.}\)

Additional exercises available at checkit.clontz.org.