Question

(a) [5pts.] Linear Independence and Linear Dependence. Determine whether the following subset S of M2x2(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.

          (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.

$(a) [5pts.] Linear Independence and Linear Dependence. Determine whether the following subset S of M2x2(R) is linearly independent or linearly dependent. S = { , , , } (b) [5pts.] Coordinates and Change of Bases. Letβ= {(1, 2, 1), (-4, 3, 4), (2, -2, -3)}Suppose the coordinate matrix ofb⃗ ∈ℝ^3relative to ordered basisβis[b⃗]β= Find[b⃗]ℰwhereℰ = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}is the standard ordered basis.$

Added by Selena G.

Question

Please give Ace some feedback