Find the matrix that corresponds to projecting onto the $y$-axis.
The transformation that projects onto the $y$-axis takes $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ to $\begin{pmatrix} 0 \\ 0 \end{pmatrix},$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ to $\begin{pmatrix} 0 \\ 1 \end{pmatrix},$ so the matrix is
\[\boxed{\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}}.\]