$v_1 = \begin{bmatrix} 1 \\ 2 \\ 1 \\ 3 \end{bmatrix}, v_2 = \begin{bmatrix} 2 \\ 4 \\ 2 \\ 6 \end{bmatrix}, v_3 = \begin{bmatrix} -5 \\ -5 \\ 0 \\ -5 \end{bmatrix}, v_4 = \begin{bmatrix} 11 \\ 15 \\ 4 \\ 19 \end{bmatrix}, v_5 = \begin{bmatrix} -3 \\ 2 \\ 5 \\ -2 \end{bmatrix}$
(1) Show that $\mathcal{B} = \{v_1, v_3, v_5\}$ is a basis for $H = \text{Span}\{v_1, v_2, v_3, v_4, v_5\}$.
(2) Consider the vector
$x = \begin{bmatrix} 8 \\ 5 \\ -3 \\ 20 \end{bmatrix}$.
Find the coordinate vector $[x]_{\mathcal{B}}$ of $x$ relative to the basis $\mathcal{B}$.
(3) Suppose $T: H \to \mathbb{R}^2$ is linear,
$T(v_1) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(v_3) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(v_5) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
Compute $T(x)$, where $x$ is the vector in the previous equation.
