I am reading Kristopher Tapp's book "Matix Groups for Undergraduates".

[note that Tapp uses the terminology $\displaystyle \mathbb{K}$ to denote one of {$\displaystyle \mathbb {R, C, H}$ where $\displaystyle \mathbb{H}$ is the set of all quaternions]

In section 5. Matrices as Linear Transformations we find the following definition:

Definition 1.10. If A $\displaystyle \in$ $\displaystyle M_n$($\displaystyle \mathbb{K}$), define $\displaystyle R_A$: $\displaystyle \mathbb{K}^n$$\displaystyle \rightarrow$ $\displaystyle \mathbb{K}^n$ such that for X$\displaystyle \in$$\displaystyle \mathbb{K}^n$,

$\displaystyle R_A$(X) := X$\displaystyle \circ$ A

Then, in section 6. The general Linear Groups we find

Definition 1.13

The general linear group over $\displaystyle \mathbb{K}$ is

$\displaystyle {GL}_n$($\displaystyle \mathbb{K}$) := {A$\displaystyle \in$$\displaystyle M_n$($\displaystyle \mathbb{K}$) | there exists B$\displaystyle \in$$\displaystyle M_n$($\displaystyle \mathbb{K}$) with AB = BA = I}

The following more visual characterization of the general linear group is often useful:

Proposition 1.14

$\displaystyle {GL}_n$($\displaystyle \mathbb{K}$) := {A$\displaystyle \in$$\displaystyle M_n$($\displaystyle \mathbb{K}$) | $\displaystyle R_A$ : $\displaystyle \mathbb{K}^n$$\displaystyle \rightarrow$$\displaystyle \mathbb{K}^n$ is a linear transformation}

Now the proof for this proposition starts as follows:

Proof:

If A $\displaystyle \in$$\displaystyle {GL}_n$($\displaystyle \mathbb{K}$) and B is such that BA = I then

$\displaystyle R_A$ $\displaystyle \circ$ $\displaystyle R_B$ $\displaystyle R_{BA}$ = $\displaystyle R_I$ = id (the identity)

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My questions are as follows:

How is $\displaystyle R_A$ $\displaystyle \circ$ $\displaystyle R_B$ = $\displaystyle R_{BA}$???

How does this work since

$\displaystyle R_A$(X) is a 1 x n matrix

$\displaystyle R_B$(X) is a 1 x n matrix

so how does $\displaystyle R_A$ $\displaystyle \circ$ $\displaystyle R_B$ does work in matrix multplication.

Futher, why does the author express the product as $\displaystyle R_{BA}$ rather than $\displaystyle R_{AB}$ {even though in this case they are equal, I am assuming that there is some point in showing the product as $\displaystyle R_{BA}$}

A final question is - what is the group operation $\displaystyle \circ$ in $\displaystyle R_A$ $\displaystyle \circ$ $\displaystyle R_B$

[I am assuming it is matrix multiplication??]

Can someone please clarify these points?

Peter