Originally Posted by
HallsofIvy Actually, the problem is not that there is not enough information but that what you give is contradictory. You say T is from $\displaystyle R^2$ to $\displaystyle R^3$ but then you say T((1, 2))= (0, 0) which is a transformation from $\displaystyle R^2$ to $\displaystyle R^3$.
Is the subspace the set of all linear transformation from $\displaystyle R^2$ to $\displaystyle R^3$ such that T((1,2))= (0,0,0)? Or did you mean that T is from $\displaystyle R^2$ to $\displaystyle R^2$?
Assuming you meant the former, any linear transformation from $\displaystyle R^2$ to $\displaystyle R^3$ can be written as a matrix:
$\displaystyle \begin{bmatrix}a & b \\ c & d \\ e & f\end{bmatrix}$
and your condition is
$\displaystyle \begin{bmatrix}a & b \\ c & d \\ e & f\end{bmatrix}\begin{bmatrix}1 \\ 2\end{bmatrix}= \begin{bmatrix}a+ 2b \\ c+ 2d \\ e+ 2f\end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
so you must have a+ 2b= 0, c+ 2d= 0, and e+ 2f= 0. Those, of course, give a= -2b, c= -2d, and e= -2f. The matrix can be written
$\displaystyle \begin{bmatrix}-2b & b \\ -2d & d \\ -2f & f\end{bmatrix}= b\begin{bmatrix}-2 & 1 \\ 0 & 0 \\ 0 & 0\end{bmatrix}+ d\begin{bmatrix}0 & 0 \\ -2 & 1 \\ 0 & 0\end{bmatrix}+ f\begin{bmatrix}0 & 0 \\ 0 & 0 \\ -2 & 1\end{bmatrix}$.
If you mean $\displaystyle R^2$ to $\displaystyle R^2$ just follow the same ideas.