Show that $P$ is an orthogonal projection matrix

Question

Let ${w_1, . . . , w_K}$ be an orthonormal basis for a subspace $W$ in $\mathbb{R}^d$ . Let $P$ be the matrix defined by

$P=\overset{K}{\underset{k=1}{\sum}}w_kw_k^T$

Show that $P$ is an orthogonal projection matrix with $\text{Range}(P) = W$.

I can show that $P=P^T=P^2$ however I'm struggling with how to show that $\text{Range}(P) = W$. I know this means that the columns of $P$ must span $W$, but how can I show this?

Answer 1

$Pw_i=\overset{K}{\underset{k=1}{\sum}}(w_kw_k^T)w_i=\overset{K}{\underset{k=1}{\sum}}w_k(w_k^Tw_i)$ and $w_k^Tw_i=1$ if $k=i$, $0$ if $k \neq i$. So $Pw_i=w_i$. Also, $Pw=0$ if $w$ is orthogonal to each $w_i$, so the range of $P$ is exactly the span of $\{w_i: 1\leq i \leq K\}$.