Let $u_1,...,u_k$ be an orthonormal basis for the subspace $W \subset \mathbb{R^m}$. Let $A = (u_1u_2...u_k)$ be the $m$ x $ k$ matrix whose columns are the orthonormal basis vectors, and define $P=AA^T$ to be the corresponding projection matrix.
a.) Given $v \in \mathbb{R^n}$, prove that its orthogonal projection $w \in W$ is given by matrix multiplication $w=Pv$
My attempt:
I know that if I let $u_1,...,u_n$ be an orthonormal basis for the subspace $W \subset V$. Then the orthogonal projection of a vector $v \in V$ onto W is $w = c_1u_1 +...+c_nu_n$ where $c_i=\langle v,u_i \rangle$ but I do not know how to show that this is given by $w=Pv$, where $W \subset \mathbb{R^m}$ and $v\in \mathbb{R^n}$?