Linear Transformations and the General Linear Group

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

[note that Tapp uses the terminology to denote one of { where is the set of all quaternions]

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

Definition 1.10. If A ( ), define : such that for X ,

(X) := X A

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

Definition 1.13

The general linear group over is

( ) := {A ( ) | there exists B ( ) with AB = BA = I}

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

Proposition 1.14

( ) := {A ( ) | : is a linear transformation}

Now the proof for this proposition starts as follows:

Proof:

If A ( ) and B is such that BA = I then

= = id (the identity)

==================================================

My questions are as follows:

How is = ???

How does this work since

(X) is a 1 x n matrix

(X) is a 1 x n matrix

so how does does work in matrix multplication.

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

A final question is - what is the group operation in

[I am assuming it is matrix multiplication??]

Can someone please clarify these points?

Peter

Re: Linear Transformations and the General Linear Group

The group operation means composition of mappings. For all X in ,

Therefore

Re: Linear Transformations and the General Linear Group

is "multiplication of the nxn matrix X on the right by the nxn matrix A". it is usually NOT the case that , because in general, and even for invertible matrices, A and B do not commute. however, if B is a (two-sided) inverse for A, then A and B DO commute.

usually, we are used to seeing the map , which does not reverse the order of composition (there are parallel left- and right- constructions for any non-commutative operation).