Real matrices with non-real eigenvalues 
I know this covers a lot, so perhaps someone could redirect me to a helpful website. 
for a) I have no idea where to start on the proof, as I don't understand why this is true.
for b) I also have no idea.
for c) A is a real matrix, so if one eigenvalue is complex, the others are too? (the conjugate of the first)?

Hint for a): What does it mean for $z$ to be an eigenvector? It means that $Az=\lambda z$. When would you say that two complex quantities are equal? You equate some parts. What, with what?
Hint for b) Assume they are linearly dependent and $y=kx$ for some $k$. Then use (a) and the fact that $b \neq 0$.
