Welcome and course outline. Review of elementary matrix and linear vector space concepts.

Welcome to the course and outline of course.

Welcome

Meaning, representations and examples of sets and spaces.

Spaces

Fields

Vector spaces

Introduction

Definition of a map (also mapping or transformation) between two sets, notations and some general properties of maps.

More examples of maps and their notations.

Examples of maps (3)

Meaning of map or mapping between two sets, domain and co-domain of a map.

Definition

Meaning of the range of a map and how it differs from its co-domain.

Range

Some examples of maps and their notations.

Examples of maps (1)

Examples of maps (2)

Meaning of the image of an element or subset of the domain for a map, and how it differs from the range of the map.

Image

Maps

Definition, examples and general properties of linear maps.

Worked examples on proof of linearity of maps.

Worked examples (1)

Properties (1)

More worked examples on proof of linearity of maps.

Worked examples (2)

Worked examples (9)

Worked examples (10)

Worked examples (11)

Worked examples (12)

Worked examples (13)

Worked examples (14)

Worked examples (15)

Properties (2)

Worked examples (3)

Worked examples (4)

Worked examples (5)

Worked examples (6)

Worked examples (7)

Worked examples (8)

Linear Maps

Analysis of the vector space of all linear maps between two given vector spaces. Addition, scalar multiplication, composition and inverse of linear maps.

Addition and scalar multiplication of linear maps and their linearity.

Addition and scalar multiplication

Vector space of all linear maps between two vector spaces over the same field of scalars - a proposition.

Vector space (1)

Vector space of all linear maps between two vector spaces over the same field of scalars - proof.

Vector space (2)

Composition of linear maps - definition and proof of its linearity.

Composition

Properties of the composition or product of linear maps - associativity, identity and distributivity.

Properties

Worked examples on composition of linear maps.

More worked examples on the composition of linear maps.

Algebra of Linear Maps

Further operations on linear maps involving their definition and properties.

Solving problems involving derivation of the definition of a linear map from some known images.

Worked examples on problems involving derivation of the definition of a linear map from some known images.

More worked examples on problems involving derivation of the definition of a linear map from some known images.

Operations on Linear Maps

Meaning of the kernel of a linear map, with worked examples.

Meaning and illustration of the kernel of a linear map.

Worked examples on kernels of linear maps.

More worked examples on kernels of linear maps.

Kernel

Meaning of the range of a linear map, with worked examples.

More worked examples on images of linear maps.

Meaning and illustration of the image of a linear map.

Worked examples on images of linear maps.

Matrix representation of a linear map, with worked examples.

Meaning and illustration of the matrix representation of a linear map.

Illustration

Worked examples on matrix representations of linear maps.

More worked examples on matrix representations of linear maps.

Matrix Representation

Meaning and properties of the transition matrix between two bases of a vector space.

More worked examples on the transition matrix between two bases of a vector space.

Meaning and illustration of the transition matrix between two bases of a vector space.

Properties of the transition matrix between two bases of a vector space.

Worked examples on the transition matrix between two bases of a vector space.

Transition Matrix

A comprehensive study of linear transformations (or maps) - definition, kernel, range, matrix representation and related matters.

Tutorials for undergraduates in engineering and physical sciences.

Linear Transformations (Maps) - Linear Algebra

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