Exercise 3.4.26

(a)

Basis for all diagonal matrices: $$\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right],\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right],\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right].$$

(b)

Add $$\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill \end{array}\right],\left[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill \end{array}\right]=$$

basis for symmetric matrices.

(c)

$$\left[\begin{array}{ccc}\hfill 0& \hfill 1& \hfill 0\\ \hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right],\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right].$$

These are simple bases (among many others) for (a) diagonal matrices (b) symmetric matrices (c) skew-symmetric matrices. The dimensions are $3,6,3$.