I need to find a base that includes only matrices with Rank=1 for the following space: $$V=\left \{ \right.(\begin{smallmatrix} x-y &2x+3y+3z \\ -14x-7y-21z & -8x+8y \end{smallmatrix}\bigr):x,y,z\in\mathbb{R}\left. \right \}$$
What I did is:
1) Extracted $x,y,x$ to find the $span$: $$V=sp\left \{ x\begin{pmatrix} 1 &2 \\ -14&8 \end{pmatrix},y\begin{pmatrix} -1 &1 \\ -7 &8 \end{pmatrix},z\begin{pmatrix} 0 &1 \\ -7 &0 \end{pmatrix} \right \}$$
2) Find the base of this $span$ using Gauss-Elimination: $$\Rightarrow V=sp\left \{ \bigl(\begin{smallmatrix} 1 &0 \\ 0 &-8 \end{smallmatrix}\bigr),\bigl(\begin{smallmatrix} 0 &1 \\ -7 &0 \end{smallmatrix}\bigr) \right \}$$
The base I found includes matrices with Rank=2 therefore doesn't comply with the question terms. I would be really glad to hear some insights about this one!