### 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.
In other words, I need a matrix that maps the standard basis vectors $\mathbf{i} = (1,0,0), \mathbf{j} = (0,1,0),$ and $\mathbf{k} = (0,0,1)$ to $(0,0,0), (0,1,0),$ and $(0,0,1)$ respectively.
This means that the columns of $\mathbf{P}$ should be $(0,0,0), (0,1,0),$ and $(0,0,1)$ in that order, since the columns of a matrix tell me how it transforms the standard basis vectors.