Linear Transformations and matrix representations

Reading Section 1.8 and 1.9

MyMathLab: Lesson U1.4

Learning Objectives

Basic

Algebraically find the image of a given vector under a linear transformation.

Given a linear transformation T(x)=Ax, find x for a given b in the image of T.

Determine if a vector is in the range of a given linear transformation.

Use linear properties to find the image of vector under a transformation.

Find the standard matrix of a linear transformation.

Demonstrate understanding of the geometric interpretation of a linear transformation.

Demonstrate understanding of one-to-one and onto properties.

Advanced

Demonstrate understanding of concepts about linear transformations and their matrices.

Geometrically describe the image of a vector under a linear transformation.

Prove that a transformation is linear or nonlinear.

Transformation, Mappings or Functions

Definitions

A transformation is (or function or mapping) from to is a rule that assigns to each vector in a vector in .

is called the domain of

is called the codomain of

The notation indicates that domain of is and that the codomain of is

For any give in the domain is a vector and is called the image of

The set of all images of is called the range of

Key Observation

Every matrix vector product is a transformation, with indicating the transformation associated with matrix A

Transformation :

The domain is

The codomain is

The range is all possible linear combinations of the columns of A, or

Example:

Lesson U1.4 Study GuideThursday, May 24, 2018 10:30 AM

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Example:

Let

,

and

and let or

Find the image of under the transformation T

Find an such that

Linear Transformations

DefinitionA transformation is a linear transformation if the following criteria hold for all in the domain of T

1) 2)

Key Observation:

Every matrix transformation is a linear transformation.

Key Observation:

If is a linear transformation, then there is a matrix such that

Key Observation:

If is a linear transformation where and are known, and , then can be computed using the linearity of the transformation

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