Let V be the real vector space of all 3×3 hermitian matrices with complex entries. Find a basis for V.
The real vector space? Then a basis consists of $\displaystyle \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$, etc. for a total of 18 "basis" matrices.
Let $\displaystyle E_{ij}$ be the matrix with a 1 in the ij position and 0 elsewhere.
Can you show
$\displaystyle E_{11}$ , $\displaystyle E_{22}$ , $\displaystyle E_{33}$
$\displaystyle E_{12}+ E_{21}$
$\displaystyle E_{13} + E_{31}$
$\displaystyle E_{23} + E_{32}$
$\displaystyle iE_{12}-iE_{21}$
$\displaystyle iE_{13}-iE_{31}$
$\displaystyle iE_{23}-iE_{32}$
form a basis?