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 $[T]_B$?