Having all real eigenvalues is apropertyof all self-adjoint linear operators but there are non-self-adjoint linear operators that also have real eigenvalues. "Explain what is meant by saying that L is self-adjoint" is asking for thedefinitionof self-adjoint. Look up the definition.

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

This makes no sense because you haven't said what the "a_ij" are. What is the questionii) a_ij= <Lvj,vi>.really?