TBIL-LA Polynomial and Matrix Spaces (VS8)

Section 2.8 Polynomial and Matrix Spaces (VS8)

Learning Outcomes

Answer questions about vector spaces of polynomials or matrices.

Subsection 2.8.1 Class Activities

Fact 2.8.1.

Every vector space with finite dimension, that is, every vector space \(V\) with a basis of the form \(\{\vec v_1,\vec v_2,\dots,\vec v_n\}\) is said to be isomorphic to a Euclidean space \(\IR^n\text{,}\) since there exists a natural correspondance between vectors in \(V\) and vectors in \(\IR^n\text{:}\)

\begin{equation*} c_1\vec v_1+c_2\vec v_2+\dots+c_n\vec v_n \leftrightarrow \left[\begin{array}{c} c_1\\c_2\\\vdots\\c_n \end{array}\right] \end{equation*}

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 2.8.3.

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 \(W\text{?}\)

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

Activity 2.8.4.

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 \(W\text{?}\)

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

Observation 2.8.5.

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.

Subsection 2.8.2 Videos

Figure 21. Video: Polynomial and matrix calculations

Subsection 2.8.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS8.slides.html.

Exercises 2.8.4 Exercises

Exercises available at https://stevenclontz.github.io/checkit-tbil-la-2021-dev/#/bank/VS8/.

Subsection 2.8.5 Mathematical Writing Explorations

Exploration 2.8.6.

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 2.8.7.

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 2.8.8.

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 2.8.6 Sample Problem and Solution

Sample problem Example B.1.12.