Learning Outcomes
- Answer questions about vector spaces of polynomials or matrices.
Section 3.6 Polynomial and Matrix Spaces (AT6)
Subsection 3.6.1 Class Activities
Observation 3.6.1.
Nearly every term we’ve defined for Euclidean vector spaces \(\mathbb R^n\) was actually defined for all kinds of vector spaces:
Activity 3.6.2.
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*}
- \(\displaystyle \left[\begin{array}{c} 0\\ 0\\ 0 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} a+b+c\\ 0\\ 0 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} a\\ b\\ c \end{array}\right]\)
Fact 3.6.3.
Every vector space with finite dimension, that is, every vector space \(V\) with a basis of the form \(\{\vec v_1,\vec v_2,\dots,\vec v_n\}\) has a linear bijection \(T\) with Euclidean space \(\IR^n\) that simply swaps its basis with the standard basis \(\{\vec e_1,\vec e_2,\dots,\vec e_n\}\) for \(\IR^n\text{:}\)
\begin{equation*}
T(c_1\vec v_1+c_2\vec v_2+\dots+c_n\vec v_n)
=
c_1\vec e_1+c_2\vec e_2+\dots+c_n\vec e_n
=
\left[\begin{array}{c}
c_1\\c_2\\\vdots\\c_n
\end{array}\right]
\end{equation*}
This transformation (in fact, any linear bijection between vector spaces) is called an isomorphism, and \(V\) is said to be isomorphic to \(\IR^n\text{.}\)
Note, in particular, that every vector space of dimension \(n\) is isomorphic to \(\IR^n\text{.}\)
Activity 3.6.4.
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 \(M_{2,2}\text{?}\)
- 2
- 3
- 4
- 5
(b)
Which Euclidean space is \(M_{2,2}\) isomorphic to?
- \(\displaystyle \IR^2\)
- \(\displaystyle \IR^3\)
- \(\displaystyle \IR^4\)
- \(\displaystyle \IR^5\)
(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 3.6.5.
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 \(\P^4\text{?}\)
- 2
- 3
- 4
- 5
(b)
Which Euclidean space is \(\P^4\) isomorphic to?
- \(\displaystyle \IR^2\)
- \(\displaystyle \IR^3\)
- \(\displaystyle \IR^4\)
- \(\displaystyle \IR^5\)
(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 3.6.6.
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 3.6.7.
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 3.6.8.
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 3.6.2 Videos
Exercises 3.6.3 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/linear-algebra/2024e/exercises/#/bank/AT6/.Subsection 3.6.4 Mathematical Writing Explorations
Exploration 3.6.9.
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.
- \(\displaystyle \left[\begin{array}{ccc}2 & 1&3\\1&-1&2\\1&0&3\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{cccc}1&-1&2&3\\3&-3&6&3\\-2&2&4&5\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc}1&3&2\\5&1&1\\6&4&3\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right]\)
Exploration 3.6.10.
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 3.6.11.
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 3.6.5 Sample Problem and Solution
Sample problem Example B.1.17.
