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?
Best Answer
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 $A$.