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Section 3.6 Polynomial and Matrix Spaces (AT6)

Subsection 3.6.1 Warm Up

Activity 3.6.1.

Consider the following vector equation and statements about it:
\begin{equation*} x_1\vec{v}_1+x_2\vec{v}_2+\cdots+x_n\vec{v}_n=\vec{w} \end{equation*}
  1. The above vector equation is consistent for every choice of \(\vec{w}\text{.}\)
  2. When the right hand is equal to \(\vec{0}\text{,}\) the equation has a unique solution.
  3. The given equation always has a unique solution, no matter what \(\vec{w}\) is.
Which, if any, of these statements make sense if we no longer assume that the vectors \(\vec{v}_1,\dots, \vec{v}_n\) are Euclidean vectors, but rather elements of a vector space?

Subsection 3.6.2 Class Activities

Activity 3.6.3.

Let \(V\) be a vector space with the basis \(\{\vec v_1,\vec v_2,\vec v_3\}\text{.}\) Which of these completes the following definition for a bijective linear map \(T:V\to\mathbb R^3\text{?}\)
\begin{equation*} T(\vec v)=T(a\vec v_1+b\vec v_2+c\vec v_3)=\left[\begin{array}{c} \unknown\\\unknown\\\unknown \end{array}\right] \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} 0\\ 0\\ 0 \end{array}\right]\)
  2. \(\displaystyle \left[\begin{array}{c} a+b+c\\ 0\\ 0 \end{array}\right]\)
  3. \(\displaystyle \left[\begin{array}{c} a\\ b\\ c \end{array}\right]\)

Activity 3.6.5.

The matrix space \(M_{2,2}=\left\{\left[\begin{array}{cc} a&b\\c&d \end{array}\right]\middle| a,b,c,d\in\IR\right\}\) has the basis
\begin{equation*} \left\{ \left[\begin{array}{cc} 1&0\\0&0 \end{array}\right], \left[\begin{array}{cc} 0&1\\0&0 \end{array}\right], \left[\begin{array}{cc} 0&0\\1&0 \end{array}\right], \left[\begin{array}{cc} 0&0\\0&1 \end{array}\right] \right\}\text{.} \end{equation*}
(a)
What is the dimension of \(M_{2,2}\text{?}\)
  1. 2
  2. 3
  3. 4
  4. 5
(b)
Which Euclidean space is \(M_{2,2}\) isomorphic to?
  1. \(\displaystyle \IR^2\)
  2. \(\displaystyle \IR^3\)
  3. \(\displaystyle \IR^4\)
  4. \(\displaystyle \IR^5\)
(c)
Describe an isomorphism \(T:M_{2,2}\to\IR^{\unknown}\text{:}\)
\begin{equation*} T\left(\left[\begin{array}{cc} a&b\\c&d \end{array}\right]\right)=\left[\begin{array}{c} \unknown\\\\\vdots\\\\\unknown \end{array}\right] \end{equation*}

Activity 3.6.6.

The polynomial space \(\P^4=\left\{a+bx+cx^2+dx^3+ex^4\middle| a,b,c,d,e\in\IR\right\}\) has the basis
\begin{equation*} \left\{1,x,x^2,x^3,x^4\right\}\text{.} \end{equation*}
(a)
What is the dimension of \(\P^4\text{?}\)
  1. 2
  2. 3
  3. 4
  4. 5
(b)
Which Euclidean space is \(\P^4\) isomorphic to?
  1. \(\displaystyle \IR^2\)
  2. \(\displaystyle \IR^3\)
  3. \(\displaystyle \IR^4\)
  4. \(\displaystyle \IR^5\)
(c)
Describe an isomorphism \(T:\P^4\to\IR^{\unknown}\text{:}\)
\begin{equation*} T\left(a+bx+cx^2+dx^3+ex^4\right)=\left[\begin{array}{c} \unknown\\\\\vdots\\\\\unknown \end{array}\right] \end{equation*}

Remark 3.6.7.

Since any finite-dimensional vector space is isomorphic to a Euclidean space \(\IR^n\text{,}\) one approach to answering questions about such spaces is to answer the corresponding question about \(\IR^n\text{.}\)

Activity 3.6.8.

Consider how to construct the polynomial \(x^3+x^2+5x+1\) as a linear combination of polynomials from the set
\begin{equation*} \left\{ x^{3} - 2 \, x^{2} + x + 2 , 2 \, x^{2} - 1 , -x^{3} + 3 \, x^{2} + 3 \, x - 2 , x^{3} - 6 \, x^{2} + 9 \, x + 5 \right\}\text{.} \end{equation*}
(a)
Describe the vector space involved in this problem, and an isomorphic Euclidean space and relevant Eucldean vectors that can be used to solve this problem.
(b)
Show how to construct an appropriate Euclidean vector from an approriate set of Euclidean vectors.
(c)
Use this result to answer the original question.

Observation 3.6.9.

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 vector spaces cannot be treated as an isomorphic finite-dimensional Euclidean space \(\IR^n\text{,}\) vectors in such vector 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.

Subsection 3.6.3 Individual Practice

Activity 3.6.10.

Let \(A=\left[\begin{array}{ccc} -2 & -1 &1\\ 1 & 0 &0\\ 0 & -4 &-2\\ 0 & 1 &3 \end{array}\right]\) and let \(T\colon\IR^3\to\IR^4\) denote the corresponding linear transformation. Note that
\begin{equation*} \RREF(A)=\left[\begin{array}{ccc} 1 & 0 &0\\ 0 & 1 &0\\ 0 & 0 &1\\ 0 & 0 &0 \end{array}\right]. \end{equation*}
The following statements are all invalid for at least one reason. Determine what makes them invalid and, suggest alternative valid statements that the author may have meant instead.
(a)
The matrix \(A\) is injective because \(\RREF(A)\) has a pivot in each column.
(b)
The matrix \(A\) does not span \(\IR^4\) because \(\RREF(A)\) has a row of zeroes.
(c)
The transformation \(T\) does not span \(\IR^4\text{.}\)
(d)
The transformation \(T\) is linearly independent.

Subsection 3.6.4 Videos

Figure 39. Video: Polynomial and matrix calculations

Exercises 3.6.5 Exercises

Subsection 3.6.6 Mathematical Writing Explorations

Exploration 3.6.11.

Given a matrix \(M\)
  • the span of the set of all columns is the column space
  • the span of the set of all rows is the row space
  • the rank of a matrix is the dimension of the column space.
Calculate the rank of these matrices.
  • \(\displaystyle \left[\begin{array}{ccc}2 & 1&3\\1&-1&2\\1&0&3\end{array}\right]\)
  • \(\displaystyle \left[\begin{array}{cccc}1&-1&2&3\\3&-3&6&3\\-2&2&4&5\end{array}\right]\)
  • \(\displaystyle \left[\begin{array}{ccc}1&3&2\\5&1&1\\6&4&3\end{array}\right]\)
  • \(\displaystyle \left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right]\)

Exploration 3.6.12.

Calculate a basis for the row space and a basis for the column space of the matrix \(\left[\begin{array}{cccc}2&0&3&4\\0&1&1&-1\\3&1&0&2\\10&-4&-1&-1\end{array}\right]\text{.}\)

Exploration 3.6.13.

If you are given the values of \(a,b,\) and \(c\text{,}\) what value of \(d\) will cause the matrix \(\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\) to have rank 1?

Subsection 3.6.7 Sample Problem and Solution

Sample problem Example B.1.17.