Eigenspaces of an orthogonal projection 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

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, $P(V)=E_1$.
