5: Linear Transformations

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5.1: 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.
5.2: 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.
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.
5.4: 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.
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.
5.6: 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.
5.7: 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.
5.8: The General Solution of a Linear System
It turns out that we can use linear transformations to solve linear systems of equations.
5.9: 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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