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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:

View attachment 9363In 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

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:

View attachment 9363In 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