7: Linear Transformations
- Page ID
- 72230
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- 7.1: 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.
- 7.2: 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.
- 7.3: 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.
- 7.4: Special Linear Transformations in R²
- In this section, we will examine some special examples of linear transformations in \(\mathbb{R}^2\) including rotations and reflections.
- 7.5: One-to-One and Onto Transformations
- This section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
- 7.7: 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.
- 7.8: The General Solution of a Linear System
- It turns out that we can use linear transformations to solve linear systems of equations.
- 7.10: 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.
- 7.11: The Kernel and Image of a Linear Map II
- Here we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.
- 7.12: The Matrix of a Linear Transformation III
- You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.
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)