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Math Help - Linear Transformations and the General Linear Group

  1. #1
    Super Member Bernhard's Avatar
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    Linear Transformations and the General Linear Group

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

    [note that Tapp uses the terminology \mathbb{K} to denote one of { \mathbb {R, C, H} where \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 \in M_n( \mathbb{K}), define R_A: \mathbb{K}^n \rightarrow \mathbb{K}^n such that for X \in \mathbb{K}^n,

    R_A(X) := X \circ A

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

    Definition 1.13

    The general linear group over \mathbb{K} is

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

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

    Proposition 1.14

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

    Now the proof for this proposition starts as follows:

    Proof:

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

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

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

    My questions are as follows:

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

    How does this work since

    R_A(X) is a 1 x n matrix
    R_B(X) is a 1 x n matrix

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

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

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

    [I am assuming it is matrix multiplication??]

    Can someone please clarify these points?

    Peter
    Last edited by Bernhard; December 25th 2011 at 07:53 PM.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Re: Linear Transformations and the General Linear Group

    The group operation \circ means composition of mappings. For all X in \mathbb{K}^n,

    (R_A\circ R_B)(X) = R_A(R_B(X)) = R_A(XB) = (XB)A = X(BA) = R_{BA}(X).

    Therefore R_A\circ R_B = R_{BA}.
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  3. #3
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    Re: Linear Transformations and the General Linear Group

    R_A(X) is "multiplication of the nxn matrix X on the right by the nxn matrix A". it is usually NOT the case that R_{BA} = R_{AB}, 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 L_A, which does not reverse the order of composition (there are parallel left- and right- constructions for any non-commutative operation).
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