UNIVERSITY OF CALIFORNIA, LOS ANGELES - SPRING 2022
MATH 115A:6 - Linear Algebra
Instructor :  Pablo S. Ocal
Office hours :  W 1:00-3:00 pm (BOELTER 8251)
Teaching Assistant :  Chuyin Jiang
Office hours :  R 2:00-3:00 pm (BOELTER 5419), R 3:00-4:00 pm (MS 2943)
Description

The course on Linear Algebra will treat, among others, proof techniques, abstract vector spaces, linear transformations between them, matrices, and inner product spaces. The prerequisites required for this course are covered in Math 33A.

The course will have weekly classes on

The book we will be following is Linear Algebra (4th edition) by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence. We will be covering material in Chapters 1, 2, 4, 5, and 6. The recommended reference books are Linear Algebra by Paul Balmer and How To Prove It by Daniel J. Velleman. Supplemental refereces are Linear Algebra Done Right by Sheldon J. Axler, Linear Algebra Done Wrong by Sergei R. Treil, and An Infinite Descent into Pure Mathematics by Clive Newstead. You may find all the information you will need on the Syllabus, and you can compute your final grade using these Grading Schemes.

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
Mar 28Sections 1.2,
Vector Spaces. Notes.
Mar 30Section 1.3,
Subspaces. Notes.
Apr 1Section 1.4, 1.5,
Linear Combinations and Systems of Linear Equations; Linear Dependence and Linear Independence. Notes.
Apr 4Section 1.5, 1.6,
Linear Dependence and Linear Independence; Bases and Dimensions. Notes.
Apr 6Section 1.6,
Bases and Dimensions. Notes.
Apr 8Section 1.6,
Bases and Dimensions. Notes.
Apr 11Section 2.1,
Linear Transformations, Null Spaces, and Ranges. Notes.
Apr 13Section 2.1,
Linear Transformations, Null Spaces, and Ranges. Notes.
Apr 15Section 2.1, 2.2
Linear Transformations, Null Spaces, and Ranges; The Matrix Representation of a Linear Transformation. Notes.
Apr 18Section 2.2,
The Matrix Representation of a Linear Transformation. Notes.
Apr 20Section 2.3,
Composition of Linear Transformations and Matrix Multiplication. Notes.
Apr 22Section 2.4,
Invertibility and Isomorphisms. Notes.
Apr 25Section 2.4, 2.5,
Invertibility and Isomorphisms; The Change of Coordinate Matrix. Notes.
Apr 27Section 2.5, 4.4,
The Change of Coordinate Matrix. Facts about Determinants. Notes.
Apr 29Section 5.1,
Eigenvalues and Eigenvectors. Notes.
May 2Sections 1.2-1.6, 2.1-2.5,
Review.
May 4Sections 1.2-1.6, 2.1-2.5,
Midterm.
May 6Section 5.1,
Eigenvalues and Eigenvectors. Notes.
May 9Section 5.2,
Diagonalizability. Notes.
May 11Section 5.2,
Diagonalizability. Notes.
May 13Section 5.2,
Diagonalizability. Notes.
May 16Section 6.1,
Inner Products and Norms. Notes.
May 18Section 6.1, 6.2,
Inner Products and Norms; The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. Notes.
May 20Section 6.2,
The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. Notes.
May 23Section 6.3,
The Adjoint of a Linear Operator. Notes.
May 25Section 6.4,
Normal and Self-Adjoint Operators. Notes.
May 27Section 6.4,
Normal and Self-Adjoint Operators. Notes.
Jun 1Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4,
Review.
Jun 3Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4,
Review.
Jun 5Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4,
Final Exam.
Homework

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 and deadlines will be posted here. You are encouraged to work in groups.

Mar 28Homework 0.
Deadline: Apr 5.
Mar 29Homework 1.
Deadline: Apr 6.
Apr 5Homework 2.
Deadline: Apr 13.
Apr 12Homework 3.
Deadline: Apr 20.
Apr 19Homework 4.
Deadline: Apr 27.
Apr 26Homework 5.
Deadline: May 4.
May 3Homework 6.
Deadline: May 11.
May 10Homework 7.
Deadline: May 18.
May 17Homework 8.
Deadline: May 25.
May 24Homework 9.
Deadline: Jun 1.
May 31Homework 10.
Deadline: Jun 8.
Discussion

There will be typically weekly discussion sessions. These are thematic worksheets 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 starred problem(s) will be due on Gradescope two days later, on Thursday at 11:59 pm. The worksheets and deadlines will be posted here. You are encouraged to work in groups.

Mar 29Worksheet 1.
Deadline: Mar 31.
Apr 5Worksheet 2.
Deadline: Apr 7.
Apr 12Worksheet 3.
Deadline: Apr 14.
Apr 19Worksheet 4.
Deadline: Apr 21.
Apr 26Worksheet 5.
Deadline: Apr 28.
May 3Worksheet 6.
Deadline: May 5.
May 10Worksheet 7.
Deadline: May 12.
May 17Worksheet 8.
Deadline: May 19.
May 24Worksheet 9.
Deadline: May 26.
May 31Worksheet 10.
Deadline: Jun 2.
Exams

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

May 4Midterm,
MS 5147 from 4:00 to 4:50 pm.
Jun 5Final Exam,
MS 5147 from 3:00 to 6:00 pm.

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

Resources

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 Benjamin Khothsombath.
  • 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.