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)

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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