I would be grateful for some help/tips/relevant website with this question.

Let (V,<,>) be a complex inner product space with an orthonormal basis {v1,v2,.......vn}. Let L:V------>V be a linear operator. Explain what is meant by saying that L is self-adjoint.

i) L is self-adjoint if and only if <Lvi,vj>=<vi,lvj>.

ii) a_ij=<Lvj,vi>.

I know if If L is self-adjoint, then all eigenvalues of L are real but how can i use this to solve i) and ii).

Thanks in advance.