Learning Outcomes
- Answer questions about vector spaces of polynomials or matrices.
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*}
- The above vector equation is consistent for every choice of \(\vec{w}\text{.}\)
- When the right hand is equal to \(\vec{0}\text{,}\) the equation has a unique solution.
- 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
Observation 3.6.2.
Nearly every term we’ve defined for Euclidean vector spaces \(\mathbb R^n\) was actually defined for all kinds of vector spaces:
Activity 3.6.3.
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)=\unknown\vec e_1+\unknown\vec e_2+\unknown\vec e_3=\left[\begin{array}{c}
\unknown\\\unknown\\\unknown
\end{array}\right]
\end{equation*}
- \(\displaystyle 0\vec e_1+0\vec e_2+0\vec e_3=\left[\begin{array}{c} 0\\ 0\\ 0 \end{array}\right]\)
- \(\displaystyle (a+b+c)\vec e_1+0\vec e_2+0\vec e_3=\left[\begin{array}{c} a+b+c\\ 0\\ 0 \end{array}\right]\)
- \(\displaystyle a\vec e_1+b\vec e_2+c\vec e_3=\left[\begin{array}{c} a\\ b\\ c \end{array}\right]\)
Fact 3.6.4.
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 3.6.5.
Consider the matrix space \(M_{2,2}=\left\{\left[\begin{array}{cc}
a&b\\c&d
\end{array}\right]\middle| a,b,c,d\in\IR\right\}\) and the following set of matrices:
\begin{equation*}
S=
\setList{\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]}.
\end{equation*}
(a)
Does the set \(S\) span \(M_{2,2}\text{?}\)
- No; the matrix \(\left[\begin{array}{cc}1&3\\2&4\end{array}\right]\) is not a linear combination of the matrices in \(S\text{.}\)
- No; the matrix \(\left[\begin{array}{cc}7&1\\0&-1\end{array}\right]\) is not a linear combination of the matrices in \(S\text{.}\)
- No; the matrix \(\left[\begin{array}{cc}-1&5\\2&9\end{array}\right]\) is not a linear combination of the matrices in \(S\text{.}\)
- Yes, every matrix in \(M_{2,2}\) is a linear combination of the matrices in \(S\text{.}\)
(b)
Is the set \(S\) linearly independent?
- No; the matrix \(\left[\begin{array}{cc}1&0\\0&0\end{array}\right]\in S\) is a linear combination of the other matrices in \(S\text{.}\)
- No; the matrix \(\left[\begin{array}{cc}0&1\\0&0\end{array}\right]\in S\) is a linear combination of the other matrices in \(S\text{.}\)
- No; the matrix \(\left[\begin{array}{cc}0&0\\1&0\end{array}\right]\in S\) is a linear combination of the other matrices in \(S\text{.}\)
- Yes; no matrix in \(S\) is a linear combination of the other matrices in \(S\text{.}\)
(c)
What statement do you think best describes the set
\begin{equation*}
S=\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\}?
\end{equation*}
- \(S\) is linearly independent
- \(S\) spans \(M_{2,2}\)
- \(S\) is a basis of \(M_{2,2}\)
- \(S\) is a basis of \(\IR^4\)
(d)
What is the dimension of \(M_{2,2}\text{?}\)
- 2
- 3
- 4
- 5
(e)
Which Euclidean space is \(M_{2,2}\) isomorphic to?
- \(\displaystyle \IR^2\)
- \(\displaystyle \IR^3\)
- \(\displaystyle \IR^4\)
- \(\displaystyle \IR^5\)
(f)
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.
Consider polynomial space \(\P^4=\left\{a+by+cy^2+dy^3+ey^4\middle| a,b,c,d,e\in\IR\right\}\) and the following set:
\begin{equation*}
S=\setList{1,y,y^2,y^3,y^4}.
\end{equation*}
(a)
Does the set \(S\) span \(\P^4\text{?}\)
- No; the polynomial \(1+y^2+2y^3\) is not a linear combination of the polynomials in \(S\text{.}\)
- No; the polynomial \(6+y-y^3+y^4\) is not a linear combination of the polynomials in \(S\text{.}\)
- No; the polynomial \(y^2+2y^3-y^4\) is not a linear combination of the polynomials in \(S\text{.}\)
- Yes; every polynomial in \(\P^4\) is a linear combination of the polynomials in \(S\text{.}\)
(b)
Is the set \(S\) linearly independent?
- No; the polynomial \(y^2\) is a linear combination of the other polynomials in \(S\text{.}\)
- No; the polynomial \(y^3\) is a linear combination of the other polynomials in \(S\text{.}\)
- No; the polynomial \(1\) is a linear combination of the other polynomials in \(S\text{.}\)
- Yes; no polynomial in \(S\) is a linear combination of the other polynomials in \(S\text{.}\)
(c)
What statement do you think best describes the set
\begin{equation*}
S=\left\{
1,y,y^2,y^3,y^4\right\}?
\end{equation*}
- \(S\) is linearly independent
- \(S\) spans \(\P^4\)
- \(S\) is a basis of \(\P^4\)
(d)
What is the dimension of \(\P^4\text{?}\)
- 2
- 3
- 4
- 5
(e)
Which Euclidean space is \(\P^4\) isomorphic to?
- \(\displaystyle \IR^2\)
- \(\displaystyle \IR^3\)
- \(\displaystyle \IR^4\)
- \(\displaystyle \IR^5\)
(f)
Describe an isomorphism \(T:\P^4\to\IR^{\unknown}\text{:}\)
\begin{equation*}
T\left(a+by+cy^2+dy^3+ey^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
Exercises 3.6.5 Exercises
Exercises available at
https://tbil.org/linear-algebra/preview/exercises/#/bank/AT6/.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.
- \(\displaystyle \left[\begin{array}{ccc}2 & 1&3\\1&-1&2\\1&0&3\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{cccc}1&-1&2&3\\3&-3&6&3\\-2&2&4&5\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc}1&3&2\\5&1&1\\6&4&3\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right]\)
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 C.1.17.
