Self-adjoint operators on $\mathbb{C}^2$

Question

I want to show that self-adjoint operators in a complex space of dimensions 2 does form a real vector space. It seems a very simple questions, but then I realize that I am not sure of how to represent these self-adjoint operators. So my question is: from where I start?

My first look whenwriting the answer was to start from the defition of a self-adjoint operator:

<Av , w > = <v , Aw >

where A is a self-adjoint operator, and v and w are vectors in C². But I am not sure if this is enough, because there is nothing there that refers specifically to C². Thus, I do not know where to start.

Answer 1

If $A$ is any self adjoint operataor and $B$ any other self adjoint opertor, what conditions on $\lambda$ and $\mu$ make $\lambda A+\mu B$ a self adjoint?