### 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: I know that the projection of a vector onto the $y$-axis is a vector that has the same $y$-component as the original vector, but has a zero $x$-component.
So, I want to find a matrix $\mathbf{P}$ that takes any vector $\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}$ and maps it to a vector $\mathbf{P} \mathbf{v} = \begin{pmatrix} 0 \\ v_y \end{pmatrix}.$
I can try to guess what the matrix $\mathbf{P}$ should look like by thinking about how it should transform the standard basis vectors $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$
I want $\mathbf{P}$ to map $\mathbf{e}_1$ to the zero vector, since $\mathbf{e}_1$ has no $y$-component.
So, the first column of $\mathbf{P}$ should be $\begin{pmatrix} 0 \\ 0 \end{pmatrix}.$
I want $\mathbf{P}$ to map $\mathbf{e}_2$ to itself, since $\mathbf{e}_2$ is already on the $y$-axis.
So, the second column of $\mathbf{P}$ should be $\begin{pmatrix} 0 \\ 1 \end{pmatrix}.$
Therefore, I can write $\mathbf{P} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$
To check if this matrix works for any vector $\mathbf{v},$ I can multiply $\mathbf{P}$ and $\mathbf{v}$ using matrix multiplication: $\mathbf{P} \mathbf{v} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} v_x \\ v_y \end{pmatrix} = \begin{pmatrix} 0 \cdot v_x + 0 \cdot v_y \\ 0 \cdot v_x + 1 \cdot v_y \end{pmatrix} = \begin{pmatrix} 0 \\ v_y \end{pmatrix},$ as expected.
So, I have found the matrix $\mathbf{P}$ that projects any vector onto the $y$-axis.