Linear Vector Spaces - Linear Algebra

Do you want to learn how to work with abstract spaces and transformations that preserve their structure? Do you want to understand the concepts of vector subspaces, linear combinations, linear dependence, basis, dimension, coordinates, and properties of vector spaces? Do you want to master the skills of defining and manipulating linear maps, their kernels, images, matrix representations, and transition matrices? If you answered yes to any of these questions, then this course is for you! Linear Algebra: Linear Vector Spaces and Linear Maps is a comprehensive and engaging course that covers the fundamentals of vector spaces and linear maps and their applications in mathematics and science. You will learn how to: - Define and classify vector spaces and their subspaces over a given scalar field - Perform operations on vectors using linear combinations and scalar multiplication - Determine whether a set of vectors is linearly dependent or independent and find a basis and dimension for a vector space or subspace - Find the coordinates of a vector with respect to a given basis and change the basis using transition matrices - Define and classify linear maps between vector spaces and find their domains, codomains, ranges, and null spaces - Find the kernel and image of a linear map and use them to determine whether a linear map is one-to-one or onto - Represent a linear map using a matrix and perform matrix operations such as addition, multiplication, and inversion - Use different methods and tools to solve systems of linear equations, such as Gaussian elimination, row reduction, and inverse matrices This course is suitable for anyone who wants to learn or review the basics of vector spaces and linear maps and their applications. It is especially useful for students and professionals in algebra, geometry, analysis, differential equations, optimization, cryptography, computer graphics, data science, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of vector spaces and linear maps and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vector spaces and linear maps. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

18

22 hrs

$ 8.58

Payment required for enrolment
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.
MTH 202: Mathematical Methods II
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

Definition of a space, linear vector space, with examples.

Chapter lessons

1.Welcome7:42

Welcome to the course and course outline.

2.Axioms, conjectures, theorems, etc.29:34

Meaning and examples of axioms, conjectures, theorems, lemma, corollary, theory, etc.

3.Spaces14:31

Meaning, representations and examples of sets and spaces.

4.Fields13:25

Meaning and examples of fields.

5.Vector spaces36:57

Definition of [linear] vector spaces.

6.n-tuples12:36

Examples of vector spaces - space of n-tuples and its operators.

7.Polynomials25:59

Examples of vector spaces - space of polynomials and its operators.

8.Matrices9:27

Examples of vector spaces - space of matrices and its operators.

9.Functions10:09

Examples of vector spaces - space of functions and its operators.

2
Vector Subspaces

Definition and examples of linear vector subspaces.

Chapter lessons

1.Definition18:13

Definition of vector subspaces.

2.Worked examples (1)1:07:32

Worked examples on linear vector subspaces.

3.Worked examples (2)44:20

More worked examples on linear vector subspaces.

4.Worked examples (3)39:52

More worked examples on linear vector subspaces.

3
Linear Combinations and Spans

Linear combinations of two or more vectors and linear spans of vector spaces.

Chapter lessons

1.Linear combination14:24

Meaning of linear combination.

2.Solution of linear equations30:21

Solving linear equations with matrices.

3.Worked examples (1)31:15

Worked examples on linear combinations of vectors in a vector space.

4.Worked examples (2)36:10

More worked examples on linear combinations of vectors in a vector space.

5.Linear span18:28

Definition of a linear span or spanning set of a vector space.

6.Worked examples (3)40:26

Worked examples on linear spans of vector spaces.

7.Worked examples (4)23:28

More worked examples on linear spans of vector spaces.

4
Linear Dependence and Independence

Linear dependence and independence of vectors in a vector space.

Chapter lessons

1.Definition18:57

Meaning of linear dependence and independence of vectors in a vector space.

2.Worked examples (1)46:21

Worked examples on linear dependence and independence of vectors in a vector space.

3.Worked examples (2)43:46

More worked examples on linear dependence and independence of vectors in a vector space.

4.Worked examples (3)28:23

More worked examples on linear dependence and independence of vectors in a vector space.

5
Basis and Dimension

Basis and dimension of linear vector spaces.

Chapter lessons

1.Definition12:52

Meaning of basis and dimension of a vector space.

2.Worked examples (1)1:19:14

Worked examples on basis and dimension of vector spaces.

3.Worked examples (2)1:23:59

More worked examples on basis and dimension of vector spaces.

4.Worked examples (3)18:05

More worked examples on basis and dimension of vector spaces.

6
Matrix Spaces

Meaning, bases and dimensions of vector spaces derived from matrices - row, column and null spaces.

Chapter lessons

1.Row space20:57

Meaning and illustration of the row space of a matrix.

2.Column space39:26

Meaning and illustration of the column space of a matrix.

3.Null space33:33

Meaning and illustration of the null space of a matrix.

7
Sums and Intersections

Sums and intersections of subspaces of a vector space - their basis, dimension and other properties.

Chapter lessons

1.Sums16:53

Meaning of the sum of two subspaces of a vector space.

2.Intersections8:43

Bases and dimensions of the sum and intersection of two subspaces of a vector space.

3.Worked examples (1)1:25:26

Worked examples on bases and dimensions of the sum and intersection of two subspaces of a vector space.

8
Coordinates

Meaning of coordinates of a vector in a vector space with respect to a given basis, and how to find the coordinates.

Chapter lessons

1.Definition17:07

Meaning of the coordinates of a vector with respect to a given basis.

2.Worked examples (1)15:54

Worked examples on the coordinates of a vector with respect to a given basis of its vector space.