Exercise 9

Suppose that $W$ is a subspace of $\mathbb{R}^{n}$ for some $n$ and suppose that $\vec{v}$ is not an element of $W$. Let the orthogonal projection of $\vec{v}$ into $W$ be the vector $\text{proj}_{W}({\vec{v}})=\vec{p}$. Show that $\vec{p}$ is the element of $W$ that is closest to $\vec{v}$.