2. Let P be the plane in $\mathbb{R}^3$ through the origin with normal vector $\vec{n} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$. Let $L: \mathbb{R}^3 \to \mathbb{R}^3$ be the linear transformation such that $L(\vec{v})$ is the projection of $\vec{v}$ onto P. Let $\vec{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}$, $\vec{v}_2 = \begin{bmatrix} 0 \\ -1 \\ 2 \end{bmatrix}$, and $\mathcal{B} = \{\vec{v}_1, \vec{v}_2, \vec{n}\}$.
(a) Show that $\{\vec{v}_1, \vec{v}_2\}$ is a basis of P.
(b) Compute the RREF of the matrix $\begin{bmatrix} -1 & 0 & 1 & | & 1 & 0 & 0 \\ 0 & -1 & 2 & | & 0 & 1 & 0 \\ 1 & 2 & 1 & | & 0 & 0 & 1 \end{bmatrix}$. Do not show your work.
(c) Using part (a) or part (b), briefly explain why you can conclude that $\mathcal{B}$ is a basis of $\mathbb{R}^3$.
(d) With $\mathcal{E}$ equal to the standard basis of $\mathbb{R}^3$, what are $[\text{id}]_\mathcal{E}$ and $\mathcal{E}[\text{id}]_\mathcal{B}$?
(e) Compute $L(\vec{v}_1)$, $L(\vec{v}_2)$, and $L(\vec{n})$.
(f) Compute $[L]_\mathcal{B}$ and $[L]_\mathcal{E}$.
