Math 2121 (Fall 2021)

Overview

Welcome to the homepage for Math 2121: Linear Algebra!

Send questions for the instructor to emarberg@ust.hk.
Send questions about grades to your TA.

Check out the syllabus here.

Lecture notes

Our lectures this semester will be online at the following times:

L1: Wednesdays & Fridays, 13:30 - 14:50, online through Zoom Meeting in Canvas
L2: Mondays & Wednesdays, 12:00 - 13:20, in LTG

New lecture notes will be posted each Friday. Try to read these before class.

Week 1:

Lecture 1: Systems of linear equations
Lecture 2: Row reduction to echelon form

Week 2:

Lecture 3: Vectors in Euclidean space
Lecture 4: Matrix-vector products, linear independence

Week 3:

Lecture 5: Linear independence, linear transformations
Lecture 6: One-to-one and onto functions

Week 4:

(No lecture on Wednesday due to holiday)
Lecture 7: Matrix operations, matrix multiplication

Week 5:

Lecture 8: Invertible functions and matrices
(No lecture on Friday/Monday due to holiday)

Week 6:

Lecture 9: Subspaces, null and column space
Lecture 10: Dimension and rank

Week 7:

Lecture 11: Introduction to determinants
Lecture 12: Properties of determinants

Week 8:

Lecture 13: Abstract vector spaces
Lecture 14: Vector spaces, eigenvectors, eigenvalues

Week 9:

Lecture 15: Eigenspaces and similarity
Lecture 16: Fibonacci numbers and diagonalization

Week 10:

Lecture 17: Complex numbers, complex eigenvalues
Lecture 18: Properties of eigenvalues

Week 11:

Lecture 19: Orthogonal vectors, orthogonal projections
Lecture 20: Orthogonal projections, Gram-Schmidt process

Week 12:

Lecture 21: Least-squares solutions
Lecture 22: Symmetric matrices

Week 13:

Lecture 23: Singular value decompositions
Lecture 24: More on SVDs

Programming tools

I will be showing some interactive Pluto notebooks in class using the Julia programming language.

Julia and Pluto installation instructions

You can view these notebooks as static HTML pages, but you are strongly encouraged to install Julia and Pluto in order to run and modify the code in each notebook yourself. This may be useful for exploring ideas, checking problems, and future reference. However, using these programming tools is completely optional and not necessary for the course.

Notebook 1: Introduction, linear systems, row operations [Pluto link]
Notebook 2: Echelon form, row reduction, solving linear systems [Pluto link]
Notebook 3: Vectors and linear combinations [Pluto link]
Notebook 4: Preview of matrix multiplication [Pluto link]
Notebook 5: Linear transformations, onto and one-to-one functions [Pluto link]
Notebook 6: Matrix multiplication and random walks [Pluto link]
Notebook 7: Geometry of invertible matrices [Pluto link]
Notebook 8: Determinants as signed volumes [Pluto link]
Notebook 9: Signs of determinants [Pluto link]
Notebook 10: Eigenvectors for the Fourier transform [Pluto link]
Notebook 11: Computing eigenvalues numerically [Pluto link]
Notebook 12: Fundamental theorem of algebra [Pluto link]
Notebook 13: Some spectral graph theory [Pluto link]
Notebook 14: Random orthogonal matrices [Pluto link]
Notebook 15: Random symmetric matrices [Pluto link]
Notebook 16: Polynomial interpolation [Pluto link]
Notebook 17: Applications of SVDs [Pluto link]

Other resources

The following is our primary textbook:

Linear Algebra and its Applications, 5th edition, by D. Lay, S. Lay, and J. McDonald

Some other online resources:

This is a great YouTube channel dedicated to linear algebra topics.
Linear Algebra Done Right by Axler is a useful supplementary textbook.
Khan academy has many good instructional videos on linear algebra.
Check out HKUST professor Jeffrey Chasnov's Matrix Algebra for Engineers on Coursera.

Grading

Grades will be computed as follows:

10%: homework assignments, weighted equally
30%: midterm examination
60%: final examination

Homework

We will have weekly homework assignments. Here are the relevant logistics:

Homework will be submitted online using WeBWorK.
Assignments will be due at midnight each Friday.
If you need extra time, be sure to request this before the deadline.
The first homework assignment will be due on 9 September.

The graded homework assignments are only a bare minimum of preparation for keeping up with the course. It is a good idea to do extra problems, such as the odd numbered exercises in the sections of the textbook corresponding to each lecture. These problems have solutions at the end of the textbook.

Midterm

We will have a 2-hour, out of class midterm.

Date: Tuesday, 26 October 2021
Time: 7:00PM-9:00PM
Location:
- LTJ (for section L1)
- Room 2303 (for tutorial T2A)
- Room 2404 (for tutorial T2B)
- Room 2405 (for tutorial T2C)
Please check your tutorial to report to the correct room.

Email the instructor right away if you need special arrangements, due to a time conflict or being unable to come to campus.

The midterm will cover Lectures 1-12 and HW 1-6.

Review problems and practice exams:

Some review problems with solutions.
Practice Midterm 1 (midterm from fall 2017) [solutions]
Practice Midterm 2 (midterm from fall 2018) [solutions]
Practice Midterm 3 (midterm from fall 2019) [solutions]

Here are solutions to the midterm.

Final

We will have a 3-hour final examination at the end of the term

Date: Saturday, 18 December 2021
Time: 12:30PM-3:30PM
Location: S H Ho Sports Hall

The format will be similar to the midterm.
The exam will be cumulative, covering all lectures and homework.

Review problems:

Some review problems with solutions.
More review problems (no solutions will be posted).
There is an optional WeBWorK assignment that goes with these problems.
If you complete this, then we will use your score to replace your lowest homework grade.

Practice exams:

Practice Exam 1 (final exam from fall 2017) [solutions]
Practice Exam 2 (final exam from fall 2018) [solutions]
Practice Exam 3 (final exam from fall 2019) is online in WeBWorK.

Here are solutions to the final exam.

Schedule

The following is a tentative course outline, with reading assignments from the textbook.

Week 1: Linear systems, row reduction to echelon form (reading: Sections 1.1-1.2)
Week 2: Vectors, matrix equations, linear independence (reading: Sections 1.3-1.5, 1.7)
Week 3: Linear independence, linear transformations (reading: Sections 1.7-1.9)
Week 4: Matrix multiplication, the inverse of a matrix (reading: Sections 2.1-2.3)
Week 5: Subspaces, bases, dimension (reading: Sections 2.4, 2.8-2.9)
Week 6: Determinants (reading: Sections 3.1-3.2)
Week 7: Vector spaces, midterm (reading: Sections 4.1-4.6)
Week 8: Eigenvectors, and eigenvalues (reading: Section 5.1)
Week 9: Similarity and diagonalisable matrices (reading: Sections 5.2-5.4)
Week 10: Complex eigenvalues, properties of eigenvalues (reading: Sections 5.5, 6.1)
Week 11: Inner products, orthogonality, and projections (reading: Sections 6.1-6.3)
Week 12: Gram-Schmidt process, least-squares problems (reading: Sections 6.4-6.6)
Week 13: Symmetric matrices, SVDs (reading: Sections 7.1, 7.4)