understanding notation for change of basis matrix with respect to two basis

Question

I have some issues understanding the following question:

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The vector space $V$ has bases $B_1 = \{v_1,v_2,v_3\}$ and $B_1' = \{v_1',v_2',v_3'\}$ and $W$ has bases $B_2 = \{w_1,w_2\}$ and $B_2' = \{w_1',w_2'\}$.

The map $T:V\rightarrow W$ is linear and satisfies

$T(v_1)=w_1'+3w_2',\ \ \ T(v_2)=2w_1'+w_2',\ \ \ T(v_3)=3w_1'+2w_2'.$

Also suppose

$v_1' = 2v_1+v_2 +3v_3,\ \ \ v_2'=v_1+5v_2+6v_3,\ \ \ v_3'=v_1+3v_2+3v_3$

and

$w_1'=3w_1+w_2,w_2'=4w_1+w_2.$

Find the matrix of $T$ with respect to $B_1$ and $B_2$ and with respect to $B_1'$ and $B_2'$

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Now here are my problems:

1. Does $[T]_{B_1}^{B_2}$ mean the $T$ matrix takes a vector from the space by $B_2$, apply both transformation and then convert it to $B_1$ coordinates or the other way around?
2. Is this question asking for $[T]^{B_1}_{B_2}$ and $[T]^{B_1'}_{B_2'}$?
3. Is it correct to summarize that " $T(B_1)=\begin{pmatrix}1&3\\2&1\\3&2\end{pmatrix}\begin{pmatrix}w_1'\\w_2'\end{pmatrix}$ " or should the order of the two matrices change?

Answer 1

1. $T_{B_1}^{B_2}$ takes the coordinates in basis $B_1$ of a vector $v\in V$ and produces the coordinates of $T(v)\in W$ in basis $B_2$.
2. It asks for the matrices $T_{B_1}^{B_2}$ and $T_{B'_1}^{B'_2}$.
3. I wouldn't write it this way, or at a pinch, I' write a line vector $(w'_1,w'_2)$ first and transpose the other matrix.

Answer 2

First of all, there exist many notations for those kind of matrices that turn a linear map into a matrix-times-vector kind of map. You should check what your prof has defined.

1. I've learned that this notation maps the coordinates of a vector in basis $B_1$ to the coordinates of a vector in basis $B_2$. So, you read from bottom to top. It's also the only option here as $T$ maps from $V$ to $W$ and $B_1$ is a basis of $V$, while $B_2$ is a basis of $W$.

2. Yes.

3. Why do you want to summarize it? I guess, you can write it that way, but that has no use in finding a solution. And the term $T(B_1)$ is normally understood as a the image set of $B_1$. Your right side looks more like a single vector, but I guess that's your way to summarize it. Nevertheless you shouldn't write it that way in an exam.