Eigenspaces of an orthogonal projection

Question

So for an orthogonal projection $P:V\rightarrow U $, the task is to find the eigenvalues and eigenspaces of P.

I have found that $\lambda = 0,1$

Then $E_0 = ker(P) = U^{\perp}$

but for $E_1$ i'm not sure

$E_1 = ker(P-Id)$

I feel like the solution for this is somewhat trivial and is staring me in the face, but i cant see it

Can anyone explain? thanks

Answer 1

Let $x \in E_1$. Then, $x=P(x) \in P(V)$. Hence, $E_1 \subseteq P(V)$.

Let, $y \in P(V)$. Then, $y=P(X)$ for some $x \in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y \in E_1$. $\therefore P(V) \subseteq E_1$.

Combining, we get, $P(V)=E_1$.

Answer 2

$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.