MATH 33A:1 - Linear Algebra and Applications
Instructor :  Pablo S. Ocal
Office hours :  M 1:00-3:00 pm (no appointment necessary)
Teaching Assistant :  Jacob Swenberg
Office hours :  T 12:00-1:00 pm (or by appointment)
Teaching Assistant :  David Popovic
Office hours :  R 12:00-1:00 pm (or by appointment)
Teaching Assistant :  Jaspreet Singh
Office hours :  T 1:00-2:00 pm (or by appointment)

The course on Linear Algebra and Applications will treat, among others, systems of linear equations, linear independence, bases and dimension, orthogonality, determinants, eigenvalues and eigenvectors, and matrix diagonalization. The prerequisites required for this course are covered in Math 3B, Math 31B, and Math 32A.

The course will have weekly classes on

The book we will be following is Linear Algebra with Applications (5th edition) by Otto Bretscher. We will be covering material in Chapters 1, 2, 3, 5, 6, 7, and 8. You may find all the information you will need on the Syllabus.

Class Notes

The following notes are a formal outline of the material we will be covering. These notes are not comprehensive nor are intended to be a substitute for the textbook or the lectures. If you find mistakes (typographical or other) in these notes, or have comments, please let me know.

The following are additional materials closely related to the class notes.

Tentative Schedule
Jan 3Sections 1.1, 1.2,
Introduction to Linear Systems, Matrices, Vectors, and Gauss-Jordan Elimination. Notes. Office hours.
Jan 5Section 1.3,
On the Solutions of Linear Systems; Matrix Algebra. Notes.
Jan 7Section 2.1,
Introduction to Linear Transformations and Their Inverses. Notes.
Jan 10Section 2.2,
Linear Transformations in Geometry. Notes. Office hours.
Jan 12Section 2.3,
Matrix Products. Notes.
Jan 14Section 2.4,
The Inverse of a Linear Transformation. Notes.
Jan 19Section 3.1,
Image and Kernel of a Linear Transformation. Notes. Office hours.
Jan 21Section 3.2,
Subspaces of ℝn; Bases and Linear Independence. Notes.
Jan 24Section 3.3,
The Dimension of a Subspace of ℝn. Notes.
Jan 26Section 3.4,
Coordinates. Notes.
Jan 28Sections 1.1-1.3, 2.1-2.4, 3.1-3.4,
Midterm 1.
Jan 31Section 5.1,
Orthogonal Projections and Orthonormal Bases. Notes. Office hours.
Feb 2Section 5.1,
Orthogonal Projections and Orthonormal Bases (continued). Notes.
Feb 4Section 5.2,
Gram-Schmidt Process and QR Factorization. Notes.
Feb 7Section 5.3,
Orthogonal Transformations and Orthogonal Matrices. Notes.
Feb 9Section 5.4,
Least squares. Notes. Office hours.
Feb 11Section 6.1,
Introduction to Determinants. Notes.
Feb 14Section 6.2,
Properties of the Determinant. Notes.
Feb 16Section 6.3,
Geometrical Interpretations of the Determinant; Cramer's Rule. Notes. Office hours.
Feb 18Sections 5.1-5.4, 6.1-6.3,
Midterm 2.
Feb 23Section 7.1,
Diagonalization. Notes.
Feb 25Section 7.2,
Finding the Eigenvalues of a Matrix. Notes.
Feb 28Section 7.3,
Finding the Eigenvectors of a Matrix. Notes.
Mar 2Section 7.5,
Complex Eigenvalues. Notes.
Mar 4Section 8.1,
Symmetric Matrices. Notes.
Mar 7Section 8.2,
Quadratic Forms. Notes.
Mar 9Section 8.3,
Singular Values. Notes. Office hours.
Mar 11Sections 1.1-1.3, 2.1-2.4, 3.1-3.4, 5.1-5.4, 6.1-6.3, 7.1-7.3, 7.5, 8.1-8.3,
Review. Notes.
Mar 12Sections 1.1-1.3, 2.1-2.4, 3.1-3.4, 5.1-5.4, 6.1-6.3, 7.1-7.3, 7.5, 8.1-8.3,
Final Exam.

There will be typically weekly homework. The assignments will be posted before Tuesdays at 9:00 am. The assignments will be due on Gradescope the following week, on Wednesday at 11:59 pm. The assignments, deadlines, and solutions will be posted here.

Jan 3Homework 0.
Deadline: Jan 11.
Jan 4Homework 1.
Deadline: Jan 12.
Jan 11Homework 2.
Deadline: Jan 19.
Jan 18Homework 3.
Deadline: Jan 26.
Jan 25Homework 4.
Deadline: Feb 2.
Feb 1Homework 5.
Deadline: Feb 9.
Feb 8Homework 6.
Deadline: Feb 16.
Feb 15Homework 7.
Deadline: Feb 23.
Feb 22Homework 8.
Deadline: Mar 2.
Mar 1Homework 9.
Deadline: Mar 9.
Mar 8Homework 10.
Deadline: Mar 16.

There will be typically weekly discussion sessions. These are thematic problems around a subject, intending to provide a deep dive into a challenging topic. The worksheets will be posted before Tuesdays at 9:00 am. The worksheets and solutions will be posted here. You are encouraged to work in groups.

Jan 4Worksheet 1, Solutions.
Jan 11Worksheet 2, Solutions.
Jan 18Worksheet 3, Solutions.
Jan 25Worksheet 4, Solutions.
Feb 1Worksheet 5, Solutions.
Feb 8Worksheet 6, Solutions.
Feb 15Worksheet 7, Solutions.
Feb 22Worksheet 8, Solutions.
Mar 1Worksheet 9, Solutions.
Mar 8Worksheet 10, Solutions.

Here the exams and their solutions will be posted. You can find the official UCLA schedule for the final exams here.

Jan 28Midterm 1, Solutions,
BUNCHE 1209B from 11:00 to 11:50 am.
Feb 18Midterm 2, Solutions,
BUNCHE 1209B from 11:00 to 11:50 am.
Mar 12Final Exam, Solutions,
TBD from 8:00 to 11:00 am.

The following are practice exams that you may use to prepare:


There are many resources available for a student at UCLA, I encourage you to use them and make the most out of them. Here are some materials specific to this course:

  • The Canvas forum can be found here. I encourage you to use this to post questions (anonymously if you wish), answers, and discuss the material presented in the lectures.
  • The open office hours at the Student Math Center can be found here.
  • The peer learning facilitator from the Academic Advancement Program is Katia Dahmani.
  • The Math Success Program from the Community Programs Office can be found here. You can go to their drop-in hours (no appointment necessary). You can also set up an appointment here.
  • The online calculator Wolfram|Alpha can be found here.

I also encourage you to form study groups with your classmates. However, please keep in mind that you will only learn how to do the exercises if you try them on your own: without trying, looking up the solutions or copying others' work is absolutely useless.