Eigenvectors/value proofs

I have a few eigenvector/value proofs and not really sure how to go about/start them, if someone could give me some ideas that would be great sorry if the notation in this post is bad, if you got any question about it i'll try to explain it a little better

show that if A is a nxn Hermitian matrix then all its eigenvalues are real

Show that any nonzero column vector v, v*v is a positive number

Show that any set of m eigenvectors of an nxn matrix A, n> (or equal) m, that correspond to distinct eigenvalues is independent