(a) [5pts.] Linear Independence and Linear Dependence. Determine whether the following subset S of M<sub>2x2</sub>(R) is linearly independent or linearly dependent.
$S = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right\}
(b) [5pts.] Coordinates and Change of Bases. Let
$\beta = \{(1, 2, 1), (-4, 3, 4), (2, -2, -3)\}$
Suppose the coordinate matrix of $\vec{b} \in \mathbb{R}^3$ relative to ordered basis $\beta$ is
$[\vec{b}]_\beta = \begin{bmatrix} 2 \\ -5 \\ -9 \end{bmatrix}$
Find $[\vec{b}]_\mathcal{E}$ where $\mathcal{E} = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ is the standard ordered basis.
