About this course: This course is all about matrices, and concisely covers the linear algebra that an engineer should know. We define matrices and how to add and multiply them, and introduce some special types of matrices. We describe the Gaussian elimination algorithm used to solve systems of linear equations and the corresponding LU decomposition of a matrix. We explain the concept of vector spaces and define the main vocabulary of linear algebra. We develop the theory of determinants and use it to solve the eigenvalue problem. After each video, there are problems to solve and I have tried to choose problems that exemplify the main idea of the lecture. I try to give enough problems for students to solidify their understanding of the material, but not so many that students feel overwhelmed and drop out. I do encourage students to attempt the given problems, but if they get stuck, full solutions can be found in the lecture notes for the course. The mathematics in this matrix algebra course is presented at the level of an advanced high school student, but typically students would take this course after completing a university-level single variable calculus course. There are no derivatives or integrals in this course, but student's are expected to have a certain level of mathematical maturity. Nevertheless, anyone who wants to learn the basics of matrix algebra is welcome to join. Lecture notes may be downloaded at https://bookboon.com/en/matrix-algebra-for-engineers-ebook or http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf Watch the course overview video at https://youtu.be/IZcyZHomFQc
Who is this class for: This course is aimed for first and second year university students interested in engineering or science. Anyone who wants to learn the basics of linear algebra with a focus on matrices is welcome to join.
Created by: The Hong Kong University of Science and Technology
Taught by: Jeffrey R. Chasnov, Professor
Department of Mathematics
| Level | Beginner |
| Commitment | 4 weeks of study, 3-4 hours/week |
| Language | English |
| How To Pass | Pass all graded assignments to complete the course. |
| User Ratings |
Syllabus
WEEK 1
MATRICES
In this week's lectures, we learn about matrices. Matrices are rectangular arrays of numbers or other mathematical objects and are fundamental to engineering mathematics. We will define matrices and how to add and multiply them, discuss some special matrices such as the identity and zero matrix, learn about transposes and inverses, and define orthogonal and permutation matrices.
11 videos, 24 readings, 4 practice quizzes
- Reading: Welcome and Course Information
- Practice Quiz: Diagnostic Quiz
- Reading: Get to Know Your Classmates
- Video: Introduction
- Video: Definition of a Matrix
- Reading: Practice: Construct Some Matrices
- Video: Addition and Multiplication of Matrices
- Reading: Practice: Matrix Addition and Multiplication
- Reading: Practice: AB=AC Does Not Imply B=C
- Reading: Practice: Matrix Multiplication Does Not Commute
- Video: Special Matrices
- Reading: Practice: AB=0 When A and B Are Not zero
- Reading: Practice: Product of Diagonal Matrices
- Reading: Practice: Product of Triangular Matrices
- Practice Quiz: Matrix Definitions
- Video: Transpose Matrix
- Reading: Practice: Transpose of a Matrix Product
- Reading: Practice: Any Square Matrix Can Be Written as the Sum of a Symmetric and Skew-Symmetric Matrix
- Reading: Practice: Construction of a Square Symmetric Matrix
- Video: Inner and Outer Products
- Reading: Practice: Example of a Symmetric Matrix
- Reading: Practice: Sum of the Squares of the Elements of a Matrix
- Video: Inverse Matrix
- Reading: Practice: Inverses of Two-by-Two Matrices
- Reading: Practice: Inverse of a Matrix Product
- Reading: Practice: Inverse of the Transpose Matrix
- Reading: Practice: Uniqueness of the Inverse
- Practice Quiz: Transposes and Inverses
- Video: Orthogonal Matrices
- Reading: Practice: Product of Orthogonal Matrices
- Reading: Practice: The Identity Matrix is Orthogonal
- Video: Orthogonal Matrices Example
- Reading: Practice: Inverse of the Rotation Matrix
- Reading: Practice: Three-dimensional Rotation
- Video: Permutation Matrices
- Reading: Practice: Three-by-Three Permutation Matrices
- Reading: Practice: Inverses of Three-by-Three Permutation Matrices
- Practice Quiz: Orthogonal Matrices
- Discussion Prompt: Week One Discussion
Graded: Week One
WEEK 2
SYSTEMS OF LINEAR EQUATIONS
In this week's lectures, we learn about solving a system of linear equations. A system of linear equations can be written in matrix form, and we can solve using Gaussian elimination. We will learn how to bring a matrix to reduced row echelon form, and how this can be used to compute a matrix inverse. We will also learn how to find the LU decomposition of a matrix, and how to use this decomposition to efficiently solve a system of linear equations.
7 videos, 6 readings, 2 practice quizzes
- Video: Gaussian Elimination
- Reading: Practice: Gaussian Elimination
- Video: Reduced Row Echelon Form
- Reading: Practice: Reduced Row Echelon Form
- Video: Computing Inverses
- Reading: Practice: Computing Inverses
- Practice Quiz: Gaussian Elimination
- Video: Elementary Matrices
- Reading: Practice: Elementary Matrices
- Video: LU Decomposition
- Reading: Practice: LU Decomposition
- Video: Solving (LU)x = b
- Reading: Practice: Solving (LU)x = b
- Practice Quiz: LU Decomposition
- Discussion Prompt: Week Two Discussion
Graded: Week Two
WEEK 3
VECTOR SPACES
In this week's lectures, we learn about vector spaces. A vector space consists of a set of vectors and a set of scalars that is closed under vector addition and scalar multiplication and that satisfies the usual rules of arithmetic. We will learn some of the vocabulary and phrases of linear algebra, such as linear independence, span, basis and dimension. We will learn about the four fundamental subspaces of a matrix, the Gram-Schmidt process, orthogonal projection, and the matrix formulation of the least-squares problem of drawing a straight line to fit noisy data.
13 videos, 14 readings, 4 practice quizzes
- Video: Vector Spaces
- Reading: Practice: Zero Vector
- Reading: Practice: Examples of Vector Spaces
- Video: Linear Independence
- Reading: Practice: Linear Independence
- Video: Span, Basis and Dimension
- Reading: Practice: Orthonormal basis
- Practice Quiz: Vector Space Definitions
- Video: Gram-Schmidt Process
- Reading: Practice: Gram-Schmidt Process
- Video: Gram-Schmidt Process Example
- Reading: Practice: Gram-Schmidt on Three-by-One Matrices
- Reading: Practice: Gram-Schmidt on Four-by-One Matrices
- Practice Quiz: Gram-Schmidt Process
- Video: Null Space
- Reading: Practice: Null Space
- Video: Application of the Null Space
- Reading: Practice: Underdetermined System of Linear Equations
- Video: Column Space
- Reading: Practice: Column Space
- Video: Row Space, Left Null Space and Rank
- Reading: Practice: Fundamental Matrix Subspaces
- Practice Quiz: Fundamental Subspaces
- Video: Orthogonal Projections
- Reading: Practice: Orthogonal Projections
- Video: The Least-Squares Problem
- Reading: Practice: Setting Up the Least-Squares Problem
- Video: Solution of the Least-Squares Problem
- Reading: Practice: Line of Best Fit
- Practice Quiz: Orthogonal Projections
- Discussion Prompt: Week Three Discussion
Graded: Week Three
WEEK 4
EIGENVALUES AND EIGENVECTORS
In this week's lectures, we will learn about determinants and the eigenvalue problem. We will learn how to compute determinants using a Laplace expansion, the Leibniz formula, or by row or column elimination. We will formulate the eigenvalue problem and learn how to find the eigenvalues and eigenvectors of a matrix. We will learn how to diagonalize a matrix using its eigenvalues and eigenvectors, and how this leads to an easy calculation of a matrix raised to a power.
13 videos, 20 readings, 3 practice quizzes
- Video: Two-by-Two and Three-by-Three Determinants
- Reading: Practice: Determinant of the Identity Matrix
- Reading: Practice: Row Interchange
- Reading: Practice: Determinant of a Matrix Product
- Video: Laplace Expansion
- Reading: Practice: Compute Determinant Using the Laplace Expansion
- Video: Leibniz Formula
- Reading: Practice: Compute Determinant Using the Leibniz Formula
- Video: Properties of a Determinant
- Reading: Practice: Determinant of a Matrix With Two Equal Rows
- Reading: Practice: Determinant is a Linear Function of Any Row
- Reading: Practice: Determinant Can Be Computed Using Row Reduction
- Reading: Practice: Compute Determinant Using Gaussian Elimination
- Practice Quiz: Determinants
- Video: The Eigenvalue Problem
- Reading: Practice: Characteristic Equation for a Three-by-Three Matrix
- Video: Finding Eigenvalues and Eigenvectors (1)
- Reading: Practice: Eigenvalues and Eigenvectors of a Two-by-Two Matrix
- Reading: Practice: Eigenvalues and Eigenvectors of a Three-by-Three Matrix
- Video: Finding Eigenvalues and Eigenvectors (2)
- Reading: Practice: Complex Eigenvalues
- Practice Quiz: The Eigenvalue Problem
- Video: Matrix Diagonalization
- Reading: Practice: Linearly Independent Eigenvectors
- Reading: Practice: Invertibility of the Eigenvector Matrix
- Video: Matrix Diagonalization Example
- Reading: Practice: Diagonalize a Three-by-Three Matrix
- Video: Powers of a Matrix
- Reading: Practice: Matrix Exponential
- Video: Powers of a Matrix Example
- Reading: Practice: Powers of a Matrix
- Practice Quiz: Matrix Diagonalization
- Discussion Prompt: Week Four Discussion
- Video: Concluding Remarks
- Reading: Please Rate this Course
- Reading: Acknowledgements
Graded: Week Four
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Ratings and Reviews
Very well-prepared and presented course on matrix/linear algebra operations, with emphasis on engineering considerations. Lecture notes with examples in PDF form are especially helpful.
good
Hands-on matrix algebra, very practical and not complex
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Excellent course, thanks so much! Really like the fact that the videos were backed by a comprehensive set lecture notes with problems AND solutions, including some proofs. This made consuming the concepts much easier. All in all a lot to swallow in this course, but great to get acquainted again (20+ years) with this subject matter. You have an excellent manner of teaching, Jeff. Thank you!