Understanding Browder's Remarks on Linear Transformations

In summary: Thanks for the help, Opalg ...In summary, Browder's book discusses differentiable maps and linear algebra. He explains how the assignment of a matrix to a linear transformation enables us to regard each element of the map as a point in Euclidean space.
  • #1
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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
 

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  • #2
Peter said:
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 \(\displaystyle \mathbb{R^{nm} }\).
 
  • #3
Opalg said:
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
 

Related to Understanding Browder's Remarks on Linear Transformations

1. What is the main concept behind Browder's Remarks on Linear Transformations?

The main concept behind Browder's Remarks on Linear Transformations is the study of linear transformations in mathematics. A linear transformation is a function that maps one vector space to another in a way that preserves the operations of addition and scalar multiplication. Browder's remarks provide insight and understanding into the properties and behaviors of linear transformations.

2. Why is understanding linear transformations important?

Understanding linear transformations is important because they are fundamental to many areas of mathematics and science, including geometry, physics, and engineering. They also have practical applications in fields such as computer graphics, data analysis, and optimization.

3. What are some key points discussed in Browder's Remarks on Linear Transformations?

Some key points discussed in Browder's Remarks on Linear Transformations include the definition and properties of linear transformations, the relationship between linear transformations and matrices, and the concept of invertibility for linear transformations. Additionally, Browder discusses the role of linear transformations in solving systems of equations and the connection to eigenvalues and eigenvectors.

4. How does Browder's Remarks on Linear Transformations relate to other areas of mathematics?

Browder's Remarks on Linear Transformations connect to other areas of mathematics such as linear algebra, abstract algebra, and functional analysis. Linear transformations are also used in differential equations, calculus, and topology. Understanding Browder's remarks can provide a foundation for further study in these areas.

5. Are there any real-world applications of linear transformations?

Yes, there are many real-world applications of linear transformations. Some examples include image and signal processing, data compression, and machine learning. Linear transformations are also used in economics, physics, and computer graphics. Understanding Browder's remarks can help in applying linear transformations to solve real-world problems.

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