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