MATH 33A:1 - Linear Algebra and Applications
Instructor :  Pablo S. Ocal
Office hours :  W 9:00-10:30 am (MS 6627)
Teaching Assistant :  Adam Zheleznyak
Office hours :  T 10:30-11:30 am (MS 2905)
Teaching Assistant :  Benjamin Goldman
Office hours :  T 9:00-10:00 am (MS 2963), R 9:00-10:00 am (MS 3974)
Teaching Assistant :  Victoria Quijano
Office hours :  T 3:15-4:15 pm (MS 3903)

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

  • Mondays, Wednesdays, and Fridays in MS, room 4000A, from 8:00 to 8:50 am.

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
Sep 29Sections 1.1, 1.2,
Introduction to Linear Systems, Matrices, Vectors, and Gauss-Jordan Elimination. Notes.
Oct 2Section 1.3,
On the Solutions of Linear Systems; Matrix Algebra. Notes.
Oct 4Section 2.1,
Introduction to Linear Transformations and Their Inverses. Notes.
Oct 6Section 2.2,
Linear Transformations in Geometry. Notes.
Oct 9Section 2.3,
Matrix Products. Notes.
Oct 11Section 2.4,
The Inverse of a Linear Transformation. Notes.
Oct 13Section 3.1,
Image and Kernel of a Linear Transformation. Notes.
Oct 16Section 3.2,
Subspaces of ℝn; Bases and Linear Independence. Notes.
Oct 18Section 3.3,
The Dimension of a Subspace of ℝn. Notes.
Oct 20Section 3.4,
Coordinates. Notes.
Oct 23Sections 1.1-1.3, 2.1-2.4, 3.1-3.4,
Oct 25Sections 1.1-1.3, 2.1-2.4, 3.1-3.4,
Midterm 1.
Oct 27Section 5.1,
Orthogonal Projections and Orthonormal Bases. Notes.
Oct 30Section 5.1,
Orthogonal Projections and Orthonormal Bases (continued). Notes.
Nov 1Section 5.2,
Gram-Schmidt Process and QR Factorization. Notes.
Nov 3Section 5.3,
Orthogonal Transformations and Orthogonal Matrices. Notes.
Nov 6Section 5.4,
Least squares. Notes.
Nov 8Section 6.1,
Introduction to Determinants. Notes.
Nov 13Section 6.2,
Properties of the Determinant. Notes.
Nov 15Section 6.3,
Geometrical Interpretations of the Determinant; Cramer's Rule. Notes.
Nov 17Sections 5.1-5.4, 6.1-6.3,
Nov 20Sections 5.1-5.4, 6.1-6.3,
Midterm 2.
Nov 22Section 7.1,
Diagonalization. Notes.
Nov 27Section 7.2,
Finding the Eigenvalues of a Matrix. Notes.
Nov 29Section 7.3,
Finding the Eigenvectors of a Matrix. Notes.
Dec 1Section 7.5,
Complex Eigenvalues. Notes.
Dec 4Section 8.1,
Symmetric Matrices. Notes.
Dec 6Section 8.2,
Quadratic Forms. Notes.
Dec 8Section 8.3,
Singular Values.
Dec 14Sections 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 Mondays at 9:00 am. The assignments will be due on Gradescope the following week, on Friday at 11:59 pm. The assignments and deadlines will be posted here. You are encouraged to work in groups.

Oct 2Homework 0.
Deadline: Oct 11.
Oct 2Homework 1.
Deadline: Oct 13.
Oct 9Homework 2.
Deadline: Oct 20.
Oct 16Homework 3.
Deadline: Oct 27.
Oct 23Homework 4.
Deadline: Nov 3.
Oct 30Homework 5.
Deadline: Nov 10.
Nov 6Homework 6.
Deadline: Nov 17.
Nov 13Homework 7.
Deadline: Dec 1.
Nov 20Homework 8.
Deadline: Dec 1.
Nov 27Homework 9.
Deadline: Dec 8.
Dec 4Homework 10.
Deadline: Dec 8.

There will be typically weekly discussion sessions. These are thematic worksheets around a subject, intending to provide a deep dive into a challenging topic while having readily available feedback. The starred problem(s) will be due on Gradescope on Thursday at 11:59 pm. The worksheets and deadlines will be posted here. You are encouraged to work in groups.

Oct 2Worksheet 1, Solutions.
Deadline: Oct 5.
Oct 9Worksheet 2, Solutions.
Deadline: Oct 12.
Oct 16Worksheet 3, Solutions.
Deadline: Oct 19.
Oct 23Worksheet 4, Solutions.
Deadline: Oct 26.
Oct 30Worksheet 5, Solutions.
Deadline: Nov 2.
Nov 6Worksheet 6, Solutions.
Deadline: Nov 9.
Nov 13Worksheet 7, Solutions.
Deadline: Nov 16.
Nov 20Worksheet 8, Solutions.
Deadline: Nov 30.
Nov 27Worksheet 9, Solutions.
Deadline: Nov 30.
Dec 4Worksheet 10, Solutions.
Deadline: Dec 7.

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

Oct 25Midterm 1, Solutions,
MS 4000A from 8:00 to 8:50 am.
Nov 20Midterm 2, Solutions,
MS 4000A from 8:00 to 8:50 am.
Dec 14Final Exam,
MS 4000A from 3:00 to 6:00 pm.

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 Emma Lu.
  • The peer learning facilitator from the UCLA Athletics Academic Excellence is Devyn Chun.
  • The Math Success Program from the Community Programs Office can be found here.
  • The Frequently Asked Questions of the Undergraduate Math Council can be found here.
  • The online calculator Wolfram|Alpha can be found here.

Here are some general materials about the Department of Mathematics, the College of Physical Sciences, and UCLA:

  • The Campus Resources document maintained by Britney Robinson can be found here.
  • The Frequently Asked Questions of the Undergraduate Math Council 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.