- #1
kostoglotov
- 234
- 6
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.
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?
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?