Clarification: this is a review problem, not a homework problem or anything—I'm not getting graded on this. That being said, I cannot seem to figure out how to do the last part.
Consider the matrix $A = u v^T$, where $u, v \in \mathbb R^{m}$.
(a) What is the rank of $A$? Find a basis for the range of $A$.
This part is easy: we know that Range($uv^T$) is just span({$u$}).
(b) List all eigenvalue of $A$. What are their geometric and algebraic multiplicities?
Since the columns of $A$ are just linear combinations of $u$, it should be easy to say that 0 is an eigenvalue with geometric and algebraic multiplicity $m - 1$ since the eigenspace associated with $\lambda = 0$ is just Null($A$). The other eigenvalue I feel can only be found through observation: $uv^T*u$ = $u<u, v>$ = $<u, v> u$, so the eigenvalue is $<u,v>$ with geometric and algebraic multiplicities 1. If there's any other way to find this, please let me know.
(c) Find the eigenvector for the nonzero eigenvalue of $A$.
From above: $u$
(d) Find an orthogonal projector onto the range of $A$.
This is pretty obvious again just by looking at the definition of a projector and because Range($A$) = span({$u$}): $\frac{1}{\|u\|^2}uu^T$.
(e) Find an orthogonal projector onto the nullspace of $A$.
No idea. I'm having a sort of disconnect here and really can't figure it out.
Hint: For $V=U\oplus W$ and respective projectors $P_{U}$ and $P_{W}$
$$I=P_{U}+P_{W}$$