Matrix Representation of Linear Transformation from R2x2 to R3

Question

We have a linear transformation T : $\mathbb R^{2\times2} \rightarrow \mathbb R^{3}$ defined by $$T\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=(a+b,2d,b).$$ Let A and B be the ordered bases for $\mathbb R^{2\times2}$ and $\mathbb R^{3}$ respectively: $$A=\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&1\\0&0\end{bmatrix},\begin{bmatrix}0&0\\1&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}\right\}$$ and $$B=\left\{(1,0,0),(0,1,0),(0,0,1)\right\}.$$ Find the matrix representation of the linear transformation $([T]_A^B)$.

It is still unclear to me how to operate on the two sets of bases to find the matrix representation of the transformation. Any help?

Answer 1

Start calculating the image of the basis $A$, for example for the first element of the basis we have $$T\left(\begin{bmatrix}1&0\\0&0\end{bmatrix}\right) = (1,0,0)$$ then write the element as a linear combination of the choosen basis for the Image. Hence the first column of the matrix representing $T$ is exactly $(1,0,0)^t$, and so on for all vectors of $A$.

Answer 2

Hint:

Note that you are searching matrix with three rows and four columns, that transforms: $$ \begin{bmatrix} a\\b\\c\\d \end{bmatrix} \to \begin{bmatrix} a+b\\2d\\b \end{bmatrix} $$

can you find this matrix?