### 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 $yz$-plane.### Response: To project a vector onto the $yz$-plane, I need to eliminate its $x$-component and keep its $y$ and $z$-components unchanged.
That means that if I multiply $\mathbf{P}$ by a standard basis vector $\mathbf{i} = (1,0,0),$ I should get $\mathbf{0} = (0,0,0).$
Similarly, if I multiply $\mathbf{P}$ by $\mathbf{j} = (0,1,0)$ or $\mathbf{k} = (0,0,1),$ I should get $\mathbf{j}$ or $\mathbf{k}$ respectively.
Therefore, the columns of $\mathbf{P}$ should be $\mathbf{0}, \mathbf{j},$ and $\mathbf{k}.$