Some questions about notation in "$[T]_\alpha^\beta$"

Question

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.

Answer 1

I agree with others that this notation isn't particularly standard, but it seems to make most sense if it's designed to work when $T:V\to W$ is a linear transformation between arbitrary finite-dimensional abstract vector spaces with bases $\alpha=(\alpha_1,\ldots,\alpha_n)$ and $\beta=(\beta_1,\ldots,\beta_m)$, respectively. (Here each $\alpha_i$ is an element of $V$, and each $\beta_i$ is an element of $W$).

The matrix $[T]_\alpha^\beta$ is then the matrix with the property that if $T(\alpha_i)=c_1\beta_1+\cdots+c_m\beta_m$, then $(c_1,\ldots,c_m)^T$ is the $i$th column of $[T]_\alpha^\beta$.

This means that if you have a vector $v\in V$ and want to find $T(v)$, then you can

1. Write $v$ as a linear combination of basis vectors from $\alpha$.
2. Collect the coefficients as a column vector $X$
3. Multiply that column vector by $[T]_\alpha^\beta$ from the left, which gives you a new column vector $Y=[T]_\alpha^\beta X$.
4. Multiply each of the elements of $Y$ with the basis vectors from $\beta$.
5. The sum $y_1\beta_1+y_2\beta_2+\cdots+y_m\beta_m$ will be the vector $T(v)\in W$.

Representing linear transformations with matrices allows transferring results from the nice, concrete setting of matrices to the more useful setting of abstract vector spaces. In particular, if we have a third vector space $Z$ with basis $\gamma$ and a linear transformation $U:W\to Z$, then function composition corresponds to matrix multiplication: $$ [U\circ T]_\alpha^\gamma = [U]_\beta^\gamma [T]_\alpha^\beta $$

Answer 2

$T:\mathbb{R}^n\to \mathbb{R}^m$ is a linear transformation, then $[T]$ is the matrix corresponding to $T$ in the standard basis $\{e_1,\dots, e_n\}$ and $\{e_1,\dots, e_m\}$. So $[T]_{\alpha}^{\beta}$ is the matrix representation of $T$ in terms of the ordered bases $\alpha$ and $\beta$. $[T]_{\alpha}^{\beta}$ is still an $m\times n$ matrix since all bases have to have the same number of elements. The subscript is the basis for the domain, while the superscript is the basis for the codomain.

Edit: The notation $[T]_{\alpha}^{\beta}$ is in no way standard notation for the matrix of $T$ in bases $\alpha$ and $\beta$ (actually, you'll find frustratingly enough that there is no agreed upon notation in linear algebra for this, especially when it comes to change of bases). However, I have seen the notation $[T]$ when talking about the matrix representation for $T$ in the standard basis elements in a few places, so I guess this is in some way standard.

Answer 3

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 $3\times2$