Matrix Mapping from R2 to R3. Determine matrix and size question?

Question

Let $T$ be a linear mapping from $\Bbb R^2$ to $\Bbb R^3$ . $T$ is represented by a matrix $A$ (‘standard matrix’). What is the size of this matrix? Determine $A$ if we know that

$$T\left(\begin{bmatrix}-1 \\ 1\end{bmatrix}\right) = \begin{bmatrix}3 \\ 0 \\ -1\end{bmatrix}\text{ and }T\left(\begin{bmatrix}-1 \\ 2\end{bmatrix}\right) = \begin{bmatrix}5 \\ 1\\0\end{bmatrix}$$

Answer 1

The columns of the standard matrix are given by $$ T\left(\begin{bmatrix} 1\\0\end{bmatrix}\right),T\left(\begin{bmatrix} 0\\1\end{bmatrix}\right) $$ Now use linearity to find $$ T\left(\begin{bmatrix} 0\\1\end{bmatrix}\right)= T\left(\begin{bmatrix} -1\\2\end{bmatrix}\right)-T\left(\begin{bmatrix} -1\\1\end{bmatrix}\right) $$ and $$ T\left(\begin{bmatrix} 1\\0\end{bmatrix}\right)= T\left(\begin{bmatrix} -1\\2\end{bmatrix}\right)-2T\left(\begin{bmatrix} -1\\1\end{bmatrix}\right) $$