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Set of all Linear Transformations, L(R^n, R^m) ... Remarks by Browder, Section 8.1, Page 179 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ...

I need some help in order to fully understand some remarks by Browder in Section 8.1, page 179 regarding the set of all linear transformations, \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) ... ...


The relevant statements by Browder follow Definition 6.10 and read as follows:



Browder - Remarks on L(R^n, R^m) ... Section 8.1, Page 179 ... .png





In the above text from Browder, we read the following:

" ... ... The assignment of a matrix to each linear transformation enables us to regard each element of \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) as a point of Euclidean space \(\displaystyle \mathbb{R^{nm} }\), and thus we can speak of open sets in \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\), of continuous functions of linear transformations, etc. ... ... "



My question is as follows:


Can someone please explain, in some detail, how/why exactly the assignment of a matrix to each linear transformation enables us to regard each element of \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) as a point of Euclidean space \(\displaystyle \mathbb{R^{nm} }\) ... ...



Help will be much appreciated ...

Peter
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,681
Can someone please explain, in some detail, how/why exactly the assignment of a matrix to each linear transformation enables us to regard each element of \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) as a point of Euclidean space \(\displaystyle \mathbb{R^{nm} }\) ... ...
A linear transformation in \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) is specified by an $m\times n$ matrix, which consists of $nm$ elements. If you string out those elements into a single row, they form the coordinates of a point in \(\displaystyle \mathbb{R^{nm} }\).
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
A linear transformation in \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) is specified by an $m\times n$ matrix, which consists of $nm$ elements. If you string out those elements into a single row, they form the coordinates of a point in \(\displaystyle \mathbb{R^{nm} }\).



Thanks for the help, Opalg ...

Peter