### 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 any vector $\mathbf{v}$ into its projection on the $y$-axis.
A projection is a linear transformation, so I can use the standard basis vectors $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ to determine the columns of $\mathbf{P}.$
If I apply $\mathbf{P}$ to $\mathbf{i},$ I should get the zero vector, since $\mathbf{i}$ has no component in the $y$-direction.
Therefore, the first column of $\mathbf{P}$ is $(0, 0).$
If I apply $\mathbf{P}$ to $\mathbf{j},$ I should get $\mathbf{j}$ itself, since $\mathbf{j}$ is already on the $y$-axis.
Therefore, the second column of $\mathbf{P}$ is $(0, 1).$
So, the matrix $\mathbf{P}$ is $\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.$