Find a matrix representing a given linear transformation

Question

$T(X) = [\{x_1-x_2+x_3\}, \{0+x_2-x_3\}, \{0+0+0\}]$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$. Find a matrix $A$ such that $T(x) = A(x)$

Can anyone point me in the right direction?

Thanks

Answer 1

Hint: write down $T(1,0,0)$, $T(0,1,0)$ and $T(0,0,1)$. What do these have to do with the columns of $A$?

Additional hint: if $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$, then

$$ A\begin{pmatrix}1 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix}a \\ d \\ g\end{pmatrix} \\ A\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix}b \\ e \\ h\end{pmatrix} \\ A\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}c \\ f \\ i\end{pmatrix} \\ $$

Compare these to $T(1,0,0)$, etc. as I suggested. (for example, $T(0,1,0) = (0 - 1 + 0, 0 + 1 - 0, 0 + 0 + 0) = (-1,1,0)$, what does this tell you about $b,e,h$?)