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4: Linear Transformations

Last updated

Dec 8, 2024
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- 3.E: Exercises
- 4.1: Linear Transformations

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4.1: Linear Transformations
Recall that when we multiply an m×n matrix by an n×1 column vector, the result is an m×1 column vector. In this section we will discuss how, through matrix multiplication, an m×n matrix transforms an n×1 column vector into an m×1 column vector.
4.2: The Matrix of a Linear Transformation I
In the above examples, the action of the linear transformations was to multiply by a matrix. It turns out that this is always the case for linear transformations.
4.3: Properties of Linear Transformations. Invertible 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.
4.4: Special Linear Transformations and Applications to Computer Graphics
In this section, we will examine some special examples of linear transformations in $\mathbb{R}^2$ including rotations and reflections.
4.5: One-to-One and Onto Transformations
This section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
4.6: Isomorphisms
A mapping $T:V\rightarrow W$ is called a linear transformation or linear map if it preserves the algebraic operations of addition and scalar multiplication.
4.7: The Kernel and Image of A Linear Map
In this section we will consider the case where the linear transformation is not necessarily an isomorphism.
4.8: The Matrix of a Linear Transformation II
We discuss the main result of this section, that is how to represent a linear transformation with respect to different bases.
4.9: The General Solution of a Linear System
It turns out that we can use linear transformations to solve linear systems of equations.
4.E: Exercises

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. (CC0; Maschen via Wikipedia)

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