# Matrix Representation of Linear Transformation from R2x2 to R3

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?

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 $$A$$.

• I have tried approaching the problem this way. We get $(1,0,0),(1,0,1),(0,0,0),(0,2,0)$. Therefore, we get a $4\times3$ matrix. Problem is, now that I have the four vectors, I can put them in a matrix, but is that enough to describe T? – WalkingPizza Oct 16 '18 at 21:04
• Yes, using the notation of @EmilioNovati, you can describe you matrix space just as any 4-dimensional space. In that case $T$ is essentialy a $3\times4$ matrix which is exactly the matrix you construct. – ALG Oct 16 '18 at 21:07
• I was unsure about the correctness of the approach because I was trying to multiply the aforementioned matrix by another one to test whether the transformation would return the same result as the multiplication, but I had switched one of the $1$s with a $0$, and the results were obviously different. Thank you! – WalkingPizza Oct 16 '18 at 21:09

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?