LESSON 4: Linear Transformations

Terminology:

transformation, domain, codomain, image, range, linear transformation, rotation, standard matrix for a linear transformation.

Objectives:

learn about linear transformations; understand why matrices are special linear transformations, learn how to find the standard matrix for a linear transformation.

Reading Assignment:

Chapter 1, Sections 1.7-1.8 (pages 66-80 only).

Lesson Outline

Key Ideas and Discussion:

A transformation T takes vectors from one set (the domain) and produces new vectors in another set (the codomain). There are many types of transformations that are useful in mathematics, but linear transformations are the most important. The key property for a linear transformation is that a linear transformation of a linear combination of vectors is a linear combination of the transformed vectors: . You might find linear transformations difficult at first, but a type of linear transformation that you are already familiar with is a matrix. If an n-vector is multiplied by an matrix A, the vector is (linearly) transformed into a m-vector. The linearity of this transformation () is one of the properties of the matrix-vector product. Some matrix transformations (like rotations and dilations) have special properties with geometric interpretations. In general, there is a (standard) matrix that can be used to represent any linear transformation from to .

Practice Problems:

1.7.1, 5, 9, 17, 19, 33 (pp. 73-75); 1.8.1, 3, 7, 13, 15, 19, 21 (pp. 83-84).

Assignment