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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?

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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 $A$.

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  • $\begingroup$ 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? $\endgroup$ – WalkingPizza Oct 16 '18 at 21:04
  • $\begingroup$ 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. $\endgroup$ – ALG Oct 16 '18 at 21:07
  • $\begingroup$ 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! $\endgroup$ – WalkingPizza Oct 16 '18 at 21:09
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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?

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