"Given the basis $B={(1;2),(-1;1)}, B1={(1;5),(-1;1)}$ and the linear transformation $T$ from $R^2 to R^2$ defined by $T(1;5)=(7;5)$ and $T(-1;1)=(3;-3)$ find the matrices $[T]BC$, $[T]B1C$, $[T]CC$ and $[T]BB $
C=standard basis
Hi, I'm having a lot of trouble with this excercise. The thing is that I can't identify the basis that I am working with, because the activity says "$T(1;5)=(7;5)$ and $T(-1;1)=(3;-3)$" but how can I know what transformation is it? I mean, is it $[T]CC$? Or $[T]B1C$? Or is it $[T]B1B1$? Because both vectors are written in the standar basis, aren't they? So that transformation must be [T]CC.
So, I need some help because I'm very confused. Obviously I don't want you to do it for me, but to explain it well to me, and if you can, do one of them as an example, please.
The transformation is always the same. Now, in order to compute, say, the matrix $[T]_B^B$, the first thing to do is to compute the images of the vectors of $B$. Since$$(1,2)=\frac12(1,5)+\frac12(-1,1),$$you have $T(1,2)=\frac12(7,5)+\frac12(3,-3)=(5,1)$. And, of course, $T(-1,1)=(3,-3)$. Now you express these vectors in the basis $B$:
So,$$[T]_B^B=\begin{bmatrix}2&0\\-3&-3\end{bmatrix}.$$The other matrices can be obtained by the same process.