Let $V$ be a real $n$-dimensional inner-product space and let $W \subset V$ be a $k$ dimensional subspace. Let $\mathbf{e}_{1}, \ldots, \mathbf{e}_{k}$ be an orthonormal basis for $W$. In terms of this basis, give a formula for the orthogonal projection $\pi: V \rightarrow W$.

Let $v \in V$. Prove that $\pi v$ is the closest point in $W$ to $v$.

[You may assume that the sequence $\mathbf{e}_{1}, \ldots, \mathbf{e}_{k}$ can be extended to an orthonormal basis $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ of $V$.]