I was wondering if anyone could help me get started on these 2 proofs. I'm having trouble getting through them. They're to help me prep for my upcoming final.

1) Let u and v be vectors in an inner product space. Prove that ||u|| = ||v|| if and only if (u + v) and (u - v) are orthogonal.

2) Let A be an (n × n) matrix, and let λ be an eigenvalue of A. Prove that if k is any scalar, then λ+k is an eigenvalue of A+kI.

Any help is greatly appreciated.