This is a follow up to a previous question, Change of Basis and Row Operations.
I'm making this concrete to make it a bit easier to write up, but I'm actually interested in the more general case.
Given a linear operator $T:R^3 \rightarrow R^2$ and ordered bases' $B=(b_1,b_2,b_3)$ and $C=(c_1, c_2)$. Let's say I have the images of the basis vectors from $B$, but I don't have any explicit formula for $T$.
Now to get the representative matrix I would just take the coordinates of each image and put them in the appropriate columns like so:
$T=[T]^B_C=\begin{bmatrix} [T(b_1)]_C & [T(b_2)]_C & [T(b_3)]_C \end{bmatrix} $
So here's the follow up question. Let's say I want to convert to new bases without calculating a change of basis matrix and applying it.
Assume that the new input basis can be easily arrived at via elementary operations on $B$. For instance $B'=(b_1+2b_3, b3, b2)$ So I should be able to take:
\begin{bmatrix} b_1 & T(b_1) \\ b_2 & T(b_2) \\ b_3 & T(b_3) \end{bmatrix}
perform row operations like so:
\begin{bmatrix} b_1+2b_3 & T(b_1)+2T(b_3) \\ b_3 & T(b_3) \\ b_2 & T(b_2) \end{bmatrix}
Now I can just transpose the right hand side and take the new coordinate vectors and I've got the new representative matrix:
$T=[T]^{B'}_C=\begin{bmatrix} [T(b_1)+2T(b_3)]_C & [T(b_3)]_C & [T(b_2)]_C \end{bmatrix} $
Also, I think I can get the explicit formula for $T$ according to the new base $B'$ by adding the rows of the row oriented matrix that I did the operations on.
So is all this generally correct? When is it a good idea?