Table of Contents | |

1. Description | |

2. Class Notes | |

3. Schedule | |

4. Homework | |

5. Discussion | |

6. Exams | |

7. Resources |

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)

Table of Contents | |

1. Description | |

2. Class Notes | |

3. Schedule | |

4. Homework | |

5. Discussion | |

6. Exams | |

7. Resources |

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

**Mondays, Wednesdays, and Fridays in MS, room 5147, from 4:00 to 4:50 pm. The online classroom is Zoom Meeting 929 6288 2718.**

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.

- Change of coordinates by
**Jas Singh**. - Essence of linear algebra by
**3Blue1Brown**. - The dimension of a vector space by
**Keith Conrad**. - The minimal polynomial and some applications by
**Keith Conrad**. - Isometries of the plane and linear algebra by
**Keith Conrad**. - Bases for infinite dimensional vector spaces by
**Karen E. Smith**.

Tentative Schedule

Mar 28 | Sections 1.2, Vector Spaces. Notes. |

Mar 30 | Section 1.3, Subspaces. Notes. |

Apr 1 | Section 1.4, 1.5, Linear Combinations and Systems of Linear Equations; Linear Dependence and Linear Independence. Notes. |

Apr 4 | Section 1.5, 1.6, Linear Dependence and Linear Independence; Bases and Dimensions. Notes. |

Apr 6 | Section 1.6, Bases and Dimensions. Notes. |

Apr 8 | Section 1.6, Bases and Dimensions. Notes. |

Apr 11 | Section 2.1, Linear Transformations, Null Spaces, and Ranges. Notes. |

Apr 13 | Section 2.1, Linear Transformations, Null Spaces, and Ranges. Notes. |

Apr 15 | Section 2.1, 2.2 Linear Transformations, Null Spaces, and Ranges; The Matrix Representation of a Linear Transformation. Notes. |

Apr 18 | Section 2.2, The Matrix Representation of a Linear Transformation. Notes. |

Apr 20 | Section 2.3, Composition of Linear Transformations and Matrix Multiplication. Notes. |

Apr 22 | Section 2.4, Invertibility and Isomorphisms. Notes. |

Apr 25 | Section 2.4, 2.5, Invertibility and Isomorphisms; The Change of Coordinate Matrix. Notes. |

Apr 27 | Section 2.5, 4.4, The Change of Coordinate Matrix. Facts about Determinants. Notes. |

Apr 29 | Section 5.1, Eigenvalues and Eigenvectors. Notes. |

May 2 | Sections 1.2-1.6, 2.1-2.5, Review. |

May 4 | Sections 1.2-1.6, 2.1-2.5, Midterm. |

May 6 | Section 5.1, Eigenvalues and Eigenvectors. Notes. |

May 9 | Section 5.2, Diagonalizability. Notes. |

May 11 | Section 5.2, Diagonalizability. Notes. |

May 13 | Section 5.2, Diagonalizability. Notes. |

May 16 | Section 6.1, Inner Products and Norms. Notes. |

May 18 | Section 6.1, 6.2, Inner Products and Norms; The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. Notes. |

May 20 | Section 6.2, The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. Notes. |

May 23 | Section 6.3, The Adjoint of a Linear Operator. Notes. |

May 25 | Section 6.4, Normal and Self-Adjoint Operators. Notes. |

May 27 | Section 6.4, Normal and Self-Adjoint Operators. Notes. |

Jun 1 | Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4, Review. |

Jun 3 | Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4, Review. |

Jun 5 | Sections 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 28 | Homework 0. Deadline: Apr 5. |

Mar 29 | Homework 1. Deadline: Apr 6. |

Apr 5 | Homework 2. Deadline: Apr 13. |

Apr 12 | Homework 3. Deadline: Apr 20. |

Apr 19 | Homework 4. Deadline: Apr 27. |

Apr 26 | Homework 5. Deadline: May 4. |

May 3 | Homework 6. Deadline: May 11. |

May 10 | Homework 7. Deadline: May 18. |

May 17 | Homework 8. Deadline: May 25. |

May 24 | Homework 9. Deadline: Jun 1. |

May 31 | Homework 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 29 | Worksheet 1. Deadline: Mar 31. |

Apr 5 | Worksheet 2. Deadline: Apr 7. |

Apr 12 | Worksheet 3. Deadline: Apr 14. |

Apr 19 | Worksheet 4. Deadline: Apr 21. |

Apr 26 | Worksheet 5. Deadline: Apr 28. |

May 3 | Worksheet 6. Deadline: May 5. |

May 10 | Worksheet 7. Deadline: May 12. |

May 17 | Worksheet 8. Deadline: May 19. |

May 24 | Worksheet 9. Deadline: May 26. |

May 31 | Worksheet 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 4 | Midterm, MS 5147 from 4:00 to 4:50 pm. |

Jun 5 | Final 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.