Question

Inner product question: Please help me out.
I have attached the question in jpg format as I'm not sure how to type it out.

Answer 1

Let V be an n-dimensional inner product space with inner product <,>. Let L : V -> V
be a linear operator with the following property:
||L(x)||=||x|| for all x element of V
(a) Prove that <L(x), L(y)> = <x.y> *hint consider L(x+y)
(b) Let B ={v1,...vn} be an orthonormal basis for V. Prove that [L]B is orthogonal.

thanks a lot. I would really appreciate it if you can explain how L(x+y) would help in this question.

Answer 2

is <L(x), L(y)> = to L(x+y) in this situation?

Answer 3

look at \(<L(x+y),L(x+y)>\)