Assume that $W$ is n-dimensional subspace of an m-dimensional vector space $V$. Find all eigenvalues and all eigenvectors of the projection operators $P_W$.
Here is my ideas:
Since $W$ is n-dimensional subspace of an m-dimensional vector space $V$, then $dim W <dim V$. Suppose $\lambda \in F $ is an eigenvalue of $P_W$. Then there exists non zero vector $w$ such as $P_W w = \lambda v $. Suppose $w \in W$ are linear independent, then $$ w = \lambda _1 w_1 + ...+ \lambda _m w_m = 0 $$
Then we have the eigenvalues of zeros, and How do i find eigen vectors?