Some questions about notation in "$[T]_\alpha^\beta$" I just have a few questions about the general meaning of the notation "$[T]_\alpha^\beta$". I would really appreciate if someone would dumb it WAY down to the most basic level (no assumptions, no leaps of logic) because most of the literature I have read on this notation is very scattered.
I want to mention that $\alpha$ and $\beta$ are the ordered bases for $R^n$ and $R^m$ respectively. $T$ is the linear transformation from $R^n \to R^m$.
Questions:


*

*What is $[T]$? 

*What are the subscript and superscript?

*Does the order of the subscript and superscript matter (which one is 
on top or bottom)?

*What are the dimensions of $[T]_\alpha^\beta$?
Thank you guys so much.

It means the matrix associated to $T$ respect the basis $V\subset\mathbb{R}^n$ and the basis $W\subset\mathbb{R}^m$
For example, if we have $T\colon\mathbb{R^2}\to\mathbb{R^3}$ such $T(x,y)=(x,x+y,y)$, and $V=\{e_1=(1,0);e_2=(0,1)\}$ a  ordered basis for $\mathbb{R^2}$ and $W=\{f_1=(1,0,0);f_2=(0,1,0);f_3=(0,0,1)\}$ a ordered basis for $\mathbb{R^2}$, then we have to do:

$T(e_1)=T(0,1)=(0,1,1)=0f_1+1f_2+1f_3~~ \Rightarrow [T(e_1)]=[0,1,1]^\intercal$
$T(e_2)=T(1,0)=(1,1,0)=1f_1+1f_2+0f_3~~ \Rightarrow [T(e_2)]=[1,1,0]^\intercal$
So $[T]_V^W=[[T(e_1)][T(e_2)]]=\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{bmatrix}$
Since $dim(V)=2$ and $dim(W)=3$, the matrix $[T]_V^W$ has order $3\times2$

