I need some clarification on this problem; my class notes and my current thought process are conflicting.
I have a linear transformation $$T(a,b) = (a+2b, 3a-b)$$ and I'm asked to find $[T]_B$ where $B = \{(1,1), (1,0)\}$.
I would answer the question by applying trhe transformation to both vectors of the matrix $B$, ofr example: $ T(1,1) = (3,2)$ and $ T(1,0) = (1,3)$ so $[T]_B$ would be $$\begin{pmatrix} 3 & 1 \\ 2 & 3\end{pmatrix}$$
In my notes for some reason, I took it a step further and converted $(3,2)$ into $2(1, 1) + 1(1,0)$ and converted $(1,3)$ into $3(1, 1) -2(1,0)$ for a final result of $$[T]_B = \begin{pmatrix} 2 & 3 \\ 1 & -2\end{pmatrix}$$ Which is the correct response for $[T]_B$?
The second one, the columns of the matrix represent the images of the basis elements as you correctly say, but you need to express the images in terms of your basis vectors. The point is, $(3,2) = T(1,1)$ with respect to the usual basis, but $(2,1) = T(1,1)$ with respect to the basis $B$ so the latter matrix is the correct one.