5: Linear Transformations — Read

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5.1: Linear Transformations
In this section we will consider how matrix multiplication transforms vectors while preserving vector addition and scalar multiplication. Such transformations are called linear transformations.
- 5.1E: Exercises for Section 5.1
5.2: The Matrix of a Linear Transformation I
In the previous section we saw that multiplication by a matrix is a linear transformation. It turns out that it is always the case that if \(T\) is any linear transformation which maps \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m},\) then it corresponds to multiplication by a matrix.
- 5.2E: Exercises for Section 5.2
5.3: Properties of Linear Transformations
Let \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation. Then there are some important properties of \(T\) which will be examined in this section.
- 5.3E: Exercises for Section 5.3
5.4: Special Linear Transformations in Two and Three Dimensions
In this section, we will examine some special examples of linear transformations in \(\mathbb{R}^2\) including rotations, reflections., and projections.
- 5.4E: Exercises for Section 5.4
5.5: One-to-One and Onto Transformations
This section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
- 5.5E: Exercises for Section 5.5
5.6: Isomorphisms
In this section we will consider linear transformations that are one to one and onto. Such linear transformations are called isomorphisms.
- 5.6E: Exercises for Section 5.6
5.7: The Kernel and Image of A Linear Map
In this section we will consider two important subspaces associated with a linear transformation: its kernel and its image.
- 5.7E: Exercises for Section 5.7
5.8: The Matrix of a Linear Transformation II
In this section we learn how to represent a linear transformation with respect to different bases.
- 5.8E: Exercises for Section 5.8
5.9: The General Solution of a Linear System
In this section we see how to use linear transformations to solve linear systems of equations.
- 5.9E: Exercises for Section 5.9

Thumbnail: A linear combination of one basis set of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis set. The linear combinations relating the first set to the other extend to a linear transformation, called the change of basis. (CC BY-SA; Maschen via Wikipedia)

This page titled 5: Linear Transformations was last modified on Thu, 03 Apr 2025 02:00:40 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Bill Wilson, De Anza College.

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