## Example 1

Write $$\begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ as a linear combination of $$\begin{bmatrix} 1 \\1 \end{bmatrix}$$ and $$\begin{bmatrix} -1 \\1 \end{bmatrix}$$.

## Example 2

Show that $$\operatorname{span}\left(\left \{ \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1-i \end{bmatrix} \right \}\right) = \mathbb{C}^2$$.

## Example 3

Is $$\begin{bmatrix} 1& 0 \\ 0 & 1\end{bmatrix}$$ in the span of $$\left \{ \begin{bmatrix} 3 & 1 \\ 1 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 2 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \right \}$$? Assume that we are working over the real numbers.

## Example 4

Let $$A,B,C \in \mathbb{R}^{2\times 2}$$ be given by $$A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$$, $$B = \begin{bmatrix} -1 & 0 \\ 1 & 0 \end{bmatrix}$$, and $$C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$. Give a description of the span of $$\{A,B,C\}$$ that is as simple as possible.

## Example 5

Consider the polynomials $$x$$ and $$x^2 - 1$$ with coefficients from the set of real numbers. Show that the span of these vectors is a proper subspace of $$P_2$$, the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$.

## Example 6

Determine if $$x^2 + 1$$ is in the span of $$\{x^2 + x + 1, x+2\}$$ where the scalars are the real numbers.

## Example 7

Let $$W$$ denote the span of $$\left\{ \begin{bmatrix} 1 \\ -1 \\ 2\end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ -1\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ -1\end{bmatrix}, \right\}$$. Show that $$W$$ is a proper subspace of $$\mathbb{R}^3$$.

## Example 8

Let $$W$$ be a subspace of $$\mathbb{R}^2$$.

1. Show that if $$W$$ contains two nonzero vectors that are not scalar multiples of each other, then $$W = \mathbb{R}^2$$.

2. Describe geometrically all the possibilities for $$W$$.