Linear Maps (Transformations) - Linear Algebra (Undergraduate Advanced)

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

34

23 hrs

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Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[OAU, Ife] MTH 202: Mathematical Methods II
[OAU, Ife] MTH 202: Mathematical Methods II
Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

1. Introduction
5

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

Chapter lessons

1-1. Welcome
10:31

Welcome to the course and outline of course.

1-2. Spaces
26:19

Meaning, representations and examples of sets and spaces.

1-3. Fields
17:22

Meaning and examples of fields.

1-4. Vector spaces
29:36

Definition of [linear] vector spaces.

1-5. Vector subspaces
15:41

Review of the definition of linear vector subspaces.

2. Maps
6

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

Chapter lessons

2-1. Definition
30:49

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

2-2. Range
10:15

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

2-3. Image
15:54

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.

2-4. Examples of maps (1)
15:41

Some examples of maps and their notations.

2-5. Examples of maps (2)
25:41

More examples of maps and their notations.

2-6. Examples of maps (3)
17:45

More examples of maps and their notations.

3. Linear Maps
3
15

Definition, examples and general properties of linear maps.

Chapter lessons

3-1. Definition
32:28

Meaning of a linear map.

3-2. Properties (1)
18:56

Some properties of linear maps.

3-3. Properties (2)
16:10

More properties of linear maps.

4. Algebra of Linear Maps
5
2

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

Chapter lessons

4-1. Addition
17:00

Addition of linear maps and its linearity.

4-2. Scalar multiplication
13:29

Scalar multiplication of linear maps and its linearity.

4-3. Vector space of linear maps
1:07:03

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

4-4. Composition
18:52

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

4-5. Properties
15:43

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

5. Operations on Linear Maps
1
3

Further operations on linear maps involving their definition and properties.

Chapter lessons

5-1. Illustration
7:04

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

6. Kernel
1
3

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

Chapter lessons

6-1. Definition
23:28

Meaning and illustration of the kernel of a linear map.

7. Range
1
3

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

Chapter lessons

7-1. Definition
24:54

Meaning and illustration of the image of a linear map.

8. Matrix Representation
3
3

Matrix representation of a linear map, with worked examples.

Chapter lessons

8-1. Theorem
32:26

Existence of matrix representation of a linear map.

8-2. Proof
37:34

Proof of the existence of matrix representation of a linear map.

8-3. Procedure
7:50

How to find the matrix representation of a linear map with respect to given bases of the domain and co-domain.

9. Transition Matrix
4
2

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

Chapter lessons

9-1. Definition
25:36

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

9-2. Change of bases (1)
23:55

How a change of the domain basis affects the matrix representation of a linear map.

9-3. Change of bases (2)
16:55

How a change of the co-domain basis affects the matrix representation of a linear map.

9-4. Change of bases (3)
18:48

How a change of bases of both the domain and co-domain affects the matrix representation of a linear map.