what is a spanning set for the vector space of a diagonal $3\times 3$ matrices? How would I determine this? Im familiar with diagonal matrices eg:
$$ \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{pmatrix} $$
but I amm unsure how I would represent this as a span.
A $3\times 3$ diagonal matrix has always the following form
$$\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 & c\end{bmatrix}.$$ where $a,b,c$ are real numbers. Now we can write it as the following sum:
$$\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 & c\end{bmatrix} = \begin{bmatrix}a & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix} + \begin{bmatrix}0 & 0 & 0\\0 & b & 0\\0 & 0 & 0\end{bmatrix} + \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & c\end{bmatrix},$$
which is just
$$\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 & c\end{bmatrix} = a\begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix} + b\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix} + c\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}.$$
This means, every $3\times 3$ diagonal matrix can be represented using the three matrices
$$\begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}. $$