Proof Relating to Projections

Question

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 $v\in \mathbb{R^n}$?

Answer 1

First of all, note that $A$ is a kind of semi orthogonal matrix and its pseudo inverse is given by $A^{\dagger}=A^{T}$. Now you are trying to find a vector which is the orthogonal projection of $v$ in the space spanned by columns of $A$. This is also the closest vector to $v$ in the euclidean norm sense. Let us call it $v_{p}$. Clearly $v_{p}=Au$ for some vector $u$ in $\mathbb{R}^{m}$. So you are trying to solve \begin{align} \min_{u ~\epsilon~ \mathbb{R}^{m} } ||v-Au||^{2} \end{align} The solution for this is clearly $u=A^{\dagger}v=A^{T}v$. Substituting this for $v_p$, we obtain $v_p=AA^{T}v=Pv$.

EDIT---

Note that, we want to solve for a $u$ such that $v=Au$ (which is equivalent to that minimization problem). Now $A^{T}A=I$, since it is orthogonal. So multiplying by $A^{T}$ on both sides of $v=Au$ gives $u=A^{T}v$, and hence the required solution.