# What subspace of 3x3 matrices is spanned by rank 1 matrices

kostoglotov
So that's the question in the text.

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

The texts answer is: all 3x3 matrices.

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

$$\begin{bmatrix}1 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\ 1 & 1 & 1\\ 0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 1 & 1 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\ 1 & 0 & 0\\ 1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\ 0 & 1 & 0\\ 0 & 1 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\ 0 & 0 & 1\\ 0 & 0 & 1\end{bmatrix}$$

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?

Homework Helper
What about the rank 1 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}.##
Sum the three of these, what do you get?
This matrix is clearly in the space of all 3x3 matrices, since any 3x3 matrix multiplied by it will still be in the space of 3x3 matrices.

Daeho Ro and kostoglotov