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I having some issues I think with actually just comprehending what the question is asking me for.

The texts answer is: all 3x3 matrices.

My answer and reasoning is:

the basis of the subspace of all rank 1 matrices is made up of the basis elements

[tex]\begin{bmatrix}1 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\ 1 & 1 & 1\\ 0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 1 & 1 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\ 1 & 0 & 0\\ 1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\ 0 & 1 & 0\\ 0 & 1 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\ 0 & 0 & 1\\ 0 & 0 & 1\end{bmatrix}[/tex]

I figure these are the minimum elements you need to create any and all rank 1 matrices. By linearly combining these matrices you can make all rank 1 matrices...why do the rank 1 matrices also span the space of all 3x3 matrices?