### Instruction: Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the $yz$-plane.### Response: I need to find a linear transformation that maps any vector $\mathbf{v}$ to a vector $\mathbf{P} \mathbf{v}$ that has the same $y$ and $z$ coordinates as $\mathbf{v}$, but zero $x$ coordinate.
That means that the first column of $\mathbf{P}$, which is the image of $\mathbf{e}_1 = (1, 0, 0)$, must be zero.
The second column of $\mathbf{P}$, which is the image of $\mathbf{e}_2 = (0, 1, 0)$, must be $\mathbf{e}_2$ itself, since $\mathbf{e}_2$ is already on the $yz$-plane.
Similarly, the third column of $\mathbf{P}$, which is the image of $\mathbf{e}_3 = (0, 0, 1)$, must be $\mathbf{e}_3$ itself.
Therefore, I can write $\mathbf{P}$ as the matrix whose columns are $\mathbf{0}, \mathbf{e}_2, \mathbf{e}_3$, or in other words, $\mathbf{P} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.