### 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 $y$-axis.### Response: To find the matrix $\mathbf{P}$, I need to figure out how it transforms the standard basis vectors $\mathbf{i}$ and $\mathbf{j}$, and then write them as the columns of $\mathbf{P}$.
If I project $\mathbf{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ onto the $y$-axis, I get the zero vector $\mathbf{0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$.
If I project $\mathbf{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ onto the $y$-axis, I get $\mathbf{j}$ itself, since it is already on the $y$-axis.
Therefore, the matrix $\mathbf{P}$ must have $\mathbf{0}$ and $\mathbf{j}$ as its columns, so I can write it as $\mathbf{P} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$.