**As Taught in**:

2020/2021

**Level**:

Undergraduate – online

**Learning Resource Types:**

=> Problem Sets

=> Notes from instructor insights

=>Reading Resources

**Course Description:**

Linear Algebra is a basic subject on matrix theory and development from introduction of linear algebra. In this “linear algebra” course, we emphasizing topics useful in other disciplines such as Physics, Economics and Social Sciences, Natural Sciences, and Engineering. This course exactly parallel the combination of theory and applications with “Introduction to Linear Algebra.”

**Course Overview:**

Linear Algebra covers matrix theory and introduction linear algebra, emphasizing topics useful in other disciplines. Linear algebra is a branch of mathematics that studies systems of linear equations and the properties of matrices. The concepts of linear algebra are extremely useful in Physics, Economics and Social Sciences, Natural Sciences, and Engineering. Due to it’s broad range of applications, linear algebra is one of the most widely taught subjects in college-level mathematics (and increasingly in high school).

**Course Format:**

Linear Algebra course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

- A complete set of
*Lecture Presentation*by Albert Tan Lie Sing. *Summary Notes*for all videos along with suggested readings.*Problem Solving Presentation*on every topic taught in this course.*Problem Sets*to do on your own with*Solutions*to compare with your submitted answers.- A selection of
*Java Demonstrations*to illustrate key concepts. - A full set of
*Problem Solving Solutions*, including review material to help you prepare.

**Prerequisites:**

Multivariable Calculus (Calculus 102)

**Text**:

The resource are assigned inside tab “Resources”

**Course Goals**:

After successfully completing the course, learner will have a good understanding the following topics and their applications:

- Systems of Linear Equations.
- Row reduction and Echelon Forms.
- Matrix operations, including Inverses.
- Block Matrices.
- Linear Dependence and Independence
- Subspaces and Bases and Dimensions.
- Orthogonal Bases and Orthogonal Projections.
- Gram-Schmidt process.
- Linear models and Least-Squares Problems.
- Determinants and their properties.
- Cramer’s Rule.
- Eigenvalues and Eigenvectors.
- Diagonalization of a Matrix.
- Symmetric Matrices.
- Positive definite Matrices.
- Similar Matrices.
- Linear Transformations.
- Singular Value Decomposition.

**Practices**

The practices are essential in learning linear algebra. They are not a test, and you are encouraged to talk to other learners about difficult problems. Especially after you have found them challenging. Talking about Linear Algebra is healthy. However, you must write your own solutions.

**Study Materials**

The textbook for this course from:

Strang, Gilbert. 2009. *Introduction to Linear Algebra 4th Ed*. Published by Wellesly-Cambridge Press. ISBN: 9780980232714

**FORMAT**

Linear Algebra course, designed for independent study, has been organized to follow the sequence of topics cover on “Introduction to Linear Algebra.” The content is organized into three major units:

- “Ax=b” and the Four Subspaces
- Least Squares, Determinants, and Eigenvalues
- Positive Definite Matrices and Applications

Each unit has been further divided into a sequence of sessions that cover an amount you might expect to complete in one sitting. Each session has a video presentation on the topic, accompanied by a lecture summary. For further study, there are suggested readings, from textbook:

Strang, Gilbert. 2009, February. *Introduction to Linear Algebra 4th Ed*. Published by: Wellesley-Cambridge Press. ISBN: 9780980232714

To help guide your learning, you will see how each problem solving is taught by an expert instructors.

Finally, within each unit you will be presented with sets of problems at strategic points, so you can test your understanding of the material.

It expected that learner will spend about 150 hours in total on this course. More than half of that time is spent preparing for class and doing assignments. Its difficult to estimate how long it will take you to complete the course, but you can probably expect to spend an hour or more working through each individual session.

**Note**:

To succeed in this course learner will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems. Linear Algebra material is presented in the first few lectures of *Calculus-102*, and again in here.

The basic operations of Linear Algebra are those learner learned in grade school – addition and multiplication to produce “Linear Combinations.” But with vectors, we move into four-dimensional space and n-dimensional space.

**A topic that will be covered are:**

*Unit 1*: “*Ax=b*” and the Four Subspaces

- The Geometry of Linear Equations
- An Overview of Key iDeas
- Elimination with Matrices
- Multiplication and Inverse Matrices
- Factorization into “A=LU”
- Transposes, Permutations, Vector Spaces
- Column Space and Nullspace
- Solving “Ax=b”: Row Reduced from “R”
- Independence, Basis and Dimension
- The Four Fundamental Subspaces
- Matrix Spaces; Rank 1; Small World Graphs
- Graphs, Networks, Incidence Matrices

*Unit 2*: Least Squares, Determinants and Eigenvalues

- Orthogonal Vectors and Subspaces
- Projections onto Subspaces
- Projection Matrices and Least Squares
- Orthogonal Matrices and Gram-Schmidt
- Properties of Determinants
- Determinant Formulas and Cofactors
- Cramer’s Rule, Inverse Matrix and Volume
- Eigenvalues and Eigenvectors
- Diagonalization and Powers of “A”
- Differential Equations and exp(At)
- Markov Matrices; Fourier Series

*Unit 3*: Positive Definite Matrices and Applications

- Symmetric Matrices and Positive Definiteness
- Complex Matrices; Fast Fourier Transform (FFT)
- Positive Definite Matrices and Minima
- Similar Matrices and Jordan Form
- Singular Value Decomposition
- Linear Transformations and their Matrices
- Change of Basis; Image Compression
- Left and Right Inverses; Pseudoinverse

*Unit 4*: Final Review

Before we start, please have an attention to the “NOTE” below.

NOTE: More material on linear algebra (and much more about differential equations) is in 2014 textbook “*Differential Equations and Linear Algebra*“. Also, in the version of 2016 textbook of “*Learn Differential Equations*.”

**Textbook**:

Strang, Gilbert. 2009. *Introduction to Linear Algebra 4th Ed. Wellesley-Cambridge Press*. ISBN: 9780980232714.

Section 1.1, 1.2, & 2.1 : The Geometry of Linear Equations

Section 1.3 : An Overview of Linear Algebra

Section 2.2 – 2.3 : Elimination with Matrices

Section 2.4 – 2.5 : Multiplication and Inverse Matrices

Section 2.6 : Factorization into “*A=LU*“

Section 2.7 : Transposes, Permutations, and Vector Spaces

Section 3.1 – 3.2 : Column Space and Nullspace

Section 3.2 : Solving “Ax=0”: Pivot Variables, Special Solutions

Section 3.3 – 3.4 : Solving “*Ax=b*“: Row Reduced from “*R*“

Section 3.5 : Independence, Basis and Dimension

Section 3.6 : The Four Fundamental Subspaces

Section 3.3 and 8.2 : Matrix Spaces – Small World Graphs

Section 8.2 : Graphs, Networks, Incidence Matrices

Chapter 1 – 3, & Section 8.2 : Problem Solving 1 Review

Section 4.1 : Orthogonal Vectors and Subspaces

Section 4.2 : Projections and Subspaces

Section 4.3 : Projection Matrices and Least Squares

Section 4.4 : Orthogonal Matrices and Gram-Schmidt

Section 5.1 : Properties of Determinants

Section 5.2 : Determinant Formulas and Cofactors

Section 5.3 : Cramer’s Rule, Inverse Matrix and Volume

Section 6.1 – 6.2 : Eigenvalues and Eigenvectors

Section 6.2 : Diagonalization and Powers of “*A*“

Section 6.3 : Differential Equations and exp(At)

Section 8.3 & 8.5 : Markov Matrices – Fourier Series

Chapter 4 – 6, Section 8.3 & 8.5 : Problem Solving 2 Review

Section 6.4 – 6.5 : Symmetric Matrices and Positive Definiteness

Section 10.2 – 10.3 : Complex Matrices Fast Fourier Transform (FFT)

Section 6.5 : Positive Definite Matrices and Minima

Section 6.6 : Similar Matrices and Jordan Form

Section 6.7 : Singular Value Decomposition

Section 7.1 : Linear Transformations and their Matrices

Section 7.2 : Change of Basis – Image Compression

Section 7.3 : Left and Right Inverses, Pseudoinverse

Chapter 6 – 7, & 10.2 – 10.3 : Problem Solving 3 Review

Resource gives users access to most of the course resources in a single location. (Please get your education login and access to each of them).

*Unit 1*: *“Ax=b”* and the Four Subspaces

**Titles**: The Geometry of Linear Equations

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: An Overview of Key Ideas

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Problem Solving: (get the link)

**Titles**: Elimination with Matrices

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Multiplication and Inverse Matrices

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Factorization into “*A=LU*“

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Transposes, Permutations, Vector Spaces

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Column Space and Nullspace

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Solving “*Ax=0*” Pivot Variables, Special Solution

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Solving “*Ax=b*” Row Reduced form “*R*“

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Independence, Basis, and Dimension

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: The Four Fundamental Subspaces

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Matrix Spaces – Small World Graphs

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Graphs, Networks, Incidence Matrices

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

*Unit 2*: Least Squares, Determinants and Eigenvalues

**Titles**: Orthogonal Vectors and Subspaces

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Projections onto Subspaces

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Projection Matrices and Least Squares

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Orthogonal Matrices and Gram-Schmidt

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Properties of Determinants

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Determinant Formulas and Cofactors

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Cramer’s Rule, Inverse Matrix and Volume

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Eigenvalues and Eigenvectors

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Diagonalization and Powers of “*A*“

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Differential Equations and exp(At)

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Markov Matrices: Fourier Series

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

*Unit 3: *Positive Definite Matrices and Applications

**Titles**: Symmetric Matrices and Positive Definiteness

Lecture Presentation: (get the link)

Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Complex Matrices Fast Fourier Transform (FFT)

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Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Positive Definite Matrices and Minima

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Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Similar Matrices and Jordan Form

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Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Singular Value and Decomposition

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Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Linear Transformations and their Matrices

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Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Change of Basis: IMage Compression

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Lecture Notes: (get the link)

Reading: check the “Readings” tab

Practices: (get the link)

Problem Solving: (get the link)

**Titles**: Left and Right Inverses; Pseudoinverse

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Lecture Notes: (get the link)

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Problem Solving: (get the link)

The practices will help you to understand more in this course “*Introduction to Linear Algebra*“. Please do login with your education username and password to start with your practices.

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