Suppose $P\subset M_n(\Bbb R^n)$ satisfies $P=P^T,P^2=P$ . Show that there exists a subspace $V$ of $\Bbb R^n$ such that $P=[\text{proj}_V]_A^A$ where $A$ represents the standard basis of $\Bbb R^n$ where the basis is taken to be the standard basis of $\Bbb R^n$.
My try:
Since $P^2=P$ so any vector $v\in \Bbb R^n$ can be uniquely represented as
$v=v-Pv+Pv$ where $v-Pv\in \ker P$ and $Pv\in \text{Image }P$
Also $P$ is identity on $Image(P)$.
But I dont understand how can I construct the set $V$ such that $P=[\text{proj}_V]_A^A$ .
Can someone please help??