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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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