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At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry"

" ... ... Each matrix \(\displaystyle A\) in this group determines an invertible linear transformation \(\displaystyle f_A: \mathbb{R} \to \mathbb{R}\) defined by \(\displaystyle f_A(x) = x A^t\) ... ... "I know that one may define entities how one wishes ... but why does Armstrong define \(\displaystyle f\) in terms of the transpose of \(\displaystyle A\) rather than just simply \(\displaystyle A\) ... there must be some reason or advantage to this ... but what is it? Can someone help to explain ...

I note in passing that Kristopher Tapp in his book, "Matrix Groups for Undergraduates" (Chapter 1, Section 5) ...

Peter=======================================================================================

The above post refers to the start of Ch. 9 of M. A. Armstrong's book, "Groups and Symmetry" ... so I am providing the relevant text ... as follows:View attachment 9568

The above post also refers to Chapter 1, Section 5 of Kristopher Tapp's book, "Matrix Groups for Undergraduates" ... so I am providing the relevant text ... as follows:View attachment 9569Note that Tapp uses \(\displaystyle \mathbb{K}\) to refer to one of the real numbers, the complex numbers or the quaternions ...Hope that helps,

Peter

*writes the following:***(see text below)**" ... ... Each matrix \(\displaystyle A\) in this group determines an invertible linear transformation \(\displaystyle f_A: \mathbb{R} \to \mathbb{R}\) defined by \(\displaystyle f_A(x) = x A^t\) ... ... "I know that one may define entities how one wishes ... but why does Armstrong define \(\displaystyle f\) in terms of the transpose of \(\displaystyle A\) rather than just simply \(\displaystyle A\) ... there must be some reason or advantage to this ... but what is it? Can someone help to explain ...

I note in passing that Kristopher Tapp in his book, "Matrix Groups for Undergraduates" (Chapter 1, Section 5) ...

*... defines the action of a linear transformation ( multiplication by a matrix \(\displaystyle A\)) as \(\displaystyle R_A = X \cdot A\) ... thus not using the transpose of \(\displaystyle A\) ...Hope that someone can help ...***see text below**Peter=======================================================================================

The above post refers to the start of Ch. 9 of M. A. Armstrong's book, "Groups and Symmetry" ... so I am providing the relevant text ... as follows:View attachment 9568

The above post also refers to Chapter 1, Section 5 of Kristopher Tapp's book, "Matrix Groups for Undergraduates" ... so I am providing the relevant text ... as follows:View attachment 9569Note that Tapp uses \(\displaystyle \mathbb{K}\) to refer to one of the real numbers, the complex numbers or the quaternions ...Hope that helps,

Peter