Matrix of a non-orthogonal projection and idempotence

Question

Can find materials with the proof for LA question. So:

Is this true that the matrix of a non-orthogonal projection is idempotent?

Answer 1

Let $V$ be a vector space which is the direct sum of subspaces $U$ and $W$, that is, each element of $V$ can be written uniquely as $u + w$, for $u \in U$ and $w\in W$.

The projection $p$ on $U$ along $W$ is the linear map $$ p(u + w) = u. $$ Now try and apply $p$ again to this, to see that the linear map $p$ is idempotent, so that its matrix with respect to any base of $V$ is idempotent.