Question

Find a basis for the subspace S of all diagonal 3  3 matrices of
R^3x3

Answer 1

Well, in general a diagonal matrix looks like this:
\[\left[\begin{matrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}\right]\]

Answer 2

So hopefully it's not clear what three matrices would form a natural basis
\[a \times vector1 + b \times vector2 + c \times vector3\]