I am looking to find vector spaces V & W and bases, ordered bases $\beta$ and $\gamma$ and a transformation $T$, Where M is the matrix that the transformation is describing.
$ M= \left[ {\begin{array}{cc} 1 & -1 & 0 & 2 \\ 0 & 3 & 1 & 1 \\ 1 & 0 & 2 & -1 \\ \end{array} } \right] $
I understand that I have quite a couple options for the my choices for V and W I chose $R^{4}$ & $R^{3}$ as my vectors spaces and let $\beta$ and $\gamma$ bet the standard bases. Now I'm confused on how to find the linear transformation associated with the matrix.
My first thought was to find the representation of a generic vector by the matrix but I'm not sure if that would be write. What I mean by this is to solve the following
$ M= \left[ {\begin{array}{cc} 1 & -1 & 0 & 2 & a_1 \\ 0 & 3 & 1 & 1 & a_2 \\ 1 & 0 & 2 & -1 & a_3\\ \end{array} } \right] $
Would this be the correct way to go about this problem or do I have it all wrong?