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2.5 Matrix Transformations

The past few sections introduced us to vectors and linear combinations as a means of thinking geometrically about the solutions to a linear system. Using matrix-vector multiplication, we rewrote a linear system as a matrix equation Ax = b and used the concepts of span and linear independence to understand when solutions exist and when they are unique.

In this section, we will explore how matrix-vector multiplication defines certain types of functions, which we call matrix transformations. In particular, we will develop some algebraic tools for thinking about matrix transformations and look at some motivating examples. In the next section, we will see how matrix transformations describe important geometric operations and how they are used in computer animation.

Reading

Try the Preview Activity and read section Chapter 2 Section 5 in Understanding Linear Algebra by Davis Austin.

A function [latex]T: \mathbb{R}^m \to \mathbb{R}^n[/latex] is called a matrix transformation if the outputs are defined by taking a matrix-vector product. In other words, if there is an [latex]n\times m[/latex] matrix, [latex]A[/latex], called the standard matrix, so that for all [latex]\vec{v} \in \mathbb{R}^m[/latex], [latex]T(\vec{v}) = A\vec{v}[/latex].

The vectors [latex]\vec{e}_1 \dots \vec{e}_m \in \mathbb{R}^m[/latex] are called the standard basis for [latex]\mathbb{R}^m[/latex]. These are unit vectors in the primary directions.

In [latex]\mathbb{R}^2, \vec{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix},[/latex] [latex]\vec{e}_2 = \begin{bmatrix} 0\\ 1 \end{bmatrix},[/latex]. These are unit vectors in the x and y directions.

In [latex]\mathbb{R}^3, \vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},[/latex] [latex]\vec{e}_2 = \begin{bmatrix} 0 \\ 1\\ 0 \end{bmatrix}, \vec{e}_3 = \begin{bmatrix} 0 \\ 0\\ 1 \end{bmatrix}[/latex]. These are unit vectors in the x, y, and z directions, which in Physics class we call [latex]\hat{i}, \hat{j}, \hat{k}[/latex].

In [latex]\mathbb{R}^4, \vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0\end{bmatrix},[/latex] [latex]\vec{e}_2 = \begin{bmatrix} 0 \\ 1\\ 0 \\0 \end{bmatrix}, \vec{e}_3 = \begin{bmatrix} 0 \\ 0\\ 1 \\ 0\end{bmatrix},\vec{e}_4 = \begin{bmatrix} 0 \\ 0\\ 0 \\ 1\end{bmatrix}[/latex].

If [latex]T: \mathbb{R}^m \to \mathbb{R}^n[/latex] is a matrix transformation, there is an [latex]n\times m[/latex] matrix, [latex]A[/latex], called the standard matrix, so that for all [latex]\vec{v} \in \mathbb{R}^m[/latex], [latex]T(\vec{v}) = A\vec{v}[/latex]. The standard matrix is given by [latex]A = [T(\vec{e}_1) \dots T(\vec{e}_m)][/latex].

 

In this video I show an example of building the matrix for a matrix transformation:

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Math 220, Matrices Copyright © 2018 by Kristen Pueschel is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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