Let $\mathbf{P}$ be the matrix that projects onto $\mathbf{j}$: that is, we want $\mathbf{P}$ to satisfy \[\mathbf{P} \mathbf{v} = \text{The projection of $\mathbf{v}$ onto } \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\]for all vectors $\mathbf{v}$. Use the pictures below to calculate \[\mathbf{P}\mathbf{i}, \mathbf{P} \mathbf{j}, \mathbf{P}\mathbf{k}\]in that order, and enter them in as columns below.
Then calculate the matrix $\mathbf{P}$ that projects onto $\mathbf{j}$.
