## Example 1

Let $$p_1 = -x+1$$, $$p_2 = x+2$$, and $$p_3 = x^2+1$$. Show that $$\{p_1,p_2,p_3\}$$ is a basis for $$P_2$$, the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$. Write $$x^2 + 2x$$ as a linear combination of $$p_1,p_2,p_3$$.

## Example 2

Let $$A = \begin{bmatrix} 1 & 1\\ 0 & 2 \\ -1 & 0 \end{bmatrix}$$ be a matrix with real entries. Is $$\begin{bmatrix} 1\\1\\1\end{bmatrix}$$ in the column space of $$A$$?

## Example 3

Let $$A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 0 & 1 & 1 & 0\\ -2 & 2 & 0 & 1 \end{bmatrix}$$. Give a basis for each of $$N(A)$$, $${\cal C}(A)$$, and $${\cal R}(A)$$.

## Example 4

Let $$A \in \mathbb{R}^{2\times 4}$$. What is the smallest possible value for the nullity of $$A$$?

## Example 5

Let $$A = \begin{bmatrix} 1 & -2 & 1 \\ -2 & 4 & -2 \\ \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 2 & 1\\ 1 & 1 & 0\\ 1 & 0 & -1 \end{bmatrix}$$ be matrices defined over the real numbers. Determine if $$N(A) = {\cal C}(B)$$.

## Example 6

Let $$A = \begin{bmatrix} 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 1\\ \end{bmatrix}$$ be defined over $$GF(2)$$. Give a basis for $$N(A)$$.

## Example 7

Let $$\mathbb{F}$$ denote a field. Let $$m$$ and $$n$$ be positive integers. Let $$A \in \mathbb{F}^{m \times n}$$. Let $$b \in \mathbb{F}^m$$. Show that $$b \in {\cal C}(A)$$ if and only if the system of linear equations $$Ax = b$$ has a solution where $$x =\begin{bmatrix} x_1\\ \vdots \\ x_n\end{bmatrix}$$.

## Example 8

Let $$\Gamma = \left ( \begin{bmatrix} 1 \\ 2\end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right )$$ be an ordered basis for $$\mathbb{R}^2$$. Let $$u = \begin{bmatrix} 0 \\ -2 \end{bmatrix}$$. What is $$[u]_{\Gamma}$$?

## Example 9

Let $$\Gamma = ( x^2-1, x+1, x)$$ be an ordered basis for the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$. Let $$u = x-1$$. What is $$[u]_{\Gamma}$$?

## Example 10

Determine the dimension of the subspace of $$\mathbb{R}^{2\times 2}$$ given by $$\left \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : \begin{array}{r} a + b + c + d = 0 \\~b - c + 2d= 0 \end{array} \right \}$$.

## Example 11

Let $$t,u,v,w \in \mathbb{R}^3$$ be given by $$t = \begin{bmatrix} 1 \\ 0 \\ -2\end{bmatrix}$$, $$u = \begin{bmatrix} 1 \\ -1\\ 2\end{bmatrix}$$, $$v = \begin{bmatrix} -3 \\ 2 \\ -2\end{bmatrix}$$, $$w = \begin{bmatrix} 2 \\ -1 \\ 0\end{bmatrix}$$. Let $$W$$ be a subspace of $$\mathbb{R}^3$$ given by the span of $$\{t,u,v,w\}$$. Show that $$W$$ is a proper subspace of $$\mathbb{R}^3$$.

## Example 12

Give a procedure for finding an orthogonal basis for $$\mathbb{R}^n$$ starting with a given nonzero vector.