Linear Transformations & Matrices: Armstrong, Tapp Chs. 9 & 1 - Explained

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In summary, M. A. Armstrong defines a linear transformation in terms of the transpose of its matrix. This allows for a more concise definition, as the map preserves multiplication.
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At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following:

" ... ... 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) ... see text below ... 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 ...

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

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Peter said:
At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following:

" ... ... 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?
The answer is given in the text from Armstrong's book that you posted. If $f_A(x)$ is defined to be $xA^T$ then the map $A\mapsto f_A$ preserves multiplication: $f_{AB} = f_Af_B$. If the transpose does not occur in the definition then the map would reverse the order and you would get $f_{AB} = f_Bf_A$.

Opalg said:
The answer is given in the text from Armstrong's book that you posted. If $f_A(x)$ is defined to be $xA^T$ then the map $A\mapsto f_A$ preserves multiplication: $f_{AB} = f_Af_B$. If the transpose does not occur in the definition then the map would reverse the order and you would get $f_{AB} = f_Bf_A$.
Thanks for the help, Opalg ...

Peter

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the operations of vector addition and scalar multiplication. In simpler terms, it is a function that takes in a vector and outputs another vector, where the output vector is a scaled or rotated version of the input vector.

2. How are linear transformations represented using matrices?

A linear transformation can be represented using a matrix by assigning each input vector's coordinates to a column vector, and the resulting output vector's coordinates to another column vector. The resulting matrix is called the transformation matrix, and it can be used to transform any vector in the input vector space to its corresponding output vector.

3. What is the relationship between linear transformations and matrices?

The relationship between linear transformations and matrices is that every linear transformation can be represented by a unique transformation matrix, and every transformation matrix corresponds to a unique linear transformation. This means that linear transformations and matrices are two different ways of representing the same mathematical concept.

4. How can linear transformations be applied to real-world problems?

Linear transformations have various applications in real-world problems, such as computer graphics, image processing, and data analysis. For example, in computer graphics, linear transformations can be used to rotate, scale, and translate objects on a screen. In image processing, they can be used to enhance or manipulate images. In data analysis, they can be used to transform data to make it easier to analyze or visualize.

5. What are some properties of linear transformations?

Some properties of linear transformations include that they preserve the origin (the zero vector), they preserve the length and direction of vectors, and they preserve parallel lines. Additionally, the composition of two linear transformations is also a linear transformation, and the inverse of a linear transformation is also a linear transformation.

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