Suppose that $V$ is a finite dimensional vector space and let $P$, an element of $L(V)$, satisfy $P^2 = P$.
Show that you can define an inner product on $V$ so that $P$ is orthogonal projection to some subspace.
I know that an idempotent projection is the direct sum of the null space and the range but I don't know what to do next. Please help.
The nullspace tells the direction of the projection, and we want it to be orthogonal to the projected space (the range).
So take any basis in both the nullspace ($e_1,\dots, e_k)$ and the range ($e_{k+1},\dots, e_n$), and define $$\big\langle \sum_i\alpha_ie_i,\ \sum_i\beta_ie_i\big\rangle \ :=\ \sum_i\alpha_i\beta_i$$ Verify that $P$ is indeed an orthogonal projection with respect to this inner product.