## Section2.8Polynomial and Matrix Spaces (VS8)

### Subsection2.8.1Class Activities

#### Observation2.8.2.

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*}

#### Activity2.8.3.

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 $$W\text{?}$$

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.

#### Activity2.8.4.

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 $$W\text{?}$$

1. The dimension of $$W$$ is 6 or less.

2. The dimension of $$W$$ is exactly 6.

3. The dimension of $$W$$ is 6 or more.

#### Observation2.8.5.

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.

### Subsection2.8.3Slideshow

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

### Exercises2.8.4Exercises

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

### Subsection2.8.5Mathematical Writing Explorations

#### Exploration2.8.6.

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.
• $$\displaystyle \left[\begin{array}{ccc}2 & 1&3\\1&-1&2\\1&0&3\end{array}\right]$$

• $$\displaystyle \left[\begin{array}{cccc}1&-1&2&3\\3&-3&6&3\\-2&2&4&5\end{array}\right]$$

• $$\displaystyle \left[\begin{array}{ccc}1&3&2\\5&1&1\\6&4&3\end{array}\right]$$

• $$\displaystyle \left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right]$$

#### Exploration2.8.7.

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

#### Exploration2.8.8.

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?

### Subsection2.8.6Sample Problem and Solution

Sample problem Example B.1.12.