Let $P$ be an orthogonal projection onto a subspace $E$ of an inner product space $V$. Dimension of $V$ and $E$ are $n,m$ respectively. Find the eigenvalues of $P$ with their multiplicities.
What I know is as follows:Since $P$ is projection implies $$ P^2=P\implies 0,1\ \text{are the eigen values.}$$ Now how do I compute their multiplicities?
You've made a good start already; you know that the eigenvalues of $P$ are $0$ and $1$, so the restriction of $P$ to the corresponding eigenspaces $E_0$ and $E_1$ is the zero map and the identity, respectively, i.e. $$P\vert_{E_0}=0_{E_0}\qquad\text{ and }\qquad P\vert_{E_1}=\operatorname{id}_{E_1}.$$ Can you tell from the 'picture' what $E_0$ and $E_1$ must be? What does that tell you about the multiplicities of the eigenvalues?