Table of Contents | |

1. Description | |

2. Class Notes | |

3. Schedule | |

4. Homework | |

5. Discussion | |

6. Exams | |

7. Resources |

UNIVERSITY OF CALIFORNIA, LOS ANGELES - SPRING 2023

MATH 115A:2 - Linear Algebra

Instructor : Pablo S. Ocal

Office hours : W 1:00-2:30 pm (MS 6118)

Teaching Assistant : Mark Kong

Office hours : TBD

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 GEOLOGY, room 6704, from 11:00 to 11:50 am.**

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**.

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.

- Proof techniques.
- Fields.
- Examples of linear transformations.
- 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

Apr 3 | Sections 1.2, Vector Spaces. Notes. |

Apr 5 | Section 1.3, Subspaces. |

Apr 7 | Section 1.4, 1.5, Linear Combinations and Systems of Linear Equations; Linear Dependence and Linear Independence. |

Apr 10 | Section 1.5, 1.6, Linear Dependence and Linear Independence; Bases and Dimensions. |

Apr 12 | Section 1.6, Bases and Dimensions. |

Apr 14 | Section 1.6, Bases and Dimensions. |

Apr 17 | Section 2.1, Linear Transformations, Null Spaces, and Ranges. Notes. |

Apr 19 | Section 2.1, Linear Transformations, Null Spaces, and Ranges. Notes. |

Apr 21 | Section 2.1, 2.2 Linear Transformations, Null Spaces, and Ranges; The Matrix Representation of a Linear Transformation. Notes. |

Apr 24 | Section 2.2, The Matrix Representation of a Linear Transformation. |

Apr 26 | Section 2.3, Composition of Linear Transformations and Matrix Multiplication. |

Apr 28 | Section 2.4, Invertibility and Isomorphisms. |

May 1 | Section 2.4, 2.5, Invertibility and Isomorphisms; The Change of Coordinate Matrix. |

May 3 | Section 2.5, 4.4, The Change of Coordinate Matrix. Facts about Determinants. |

May 5 | Section 5.1, Eigenvalues and Eigenvectors. |

May 8 | Sections 1.2-1.6, 2.1-2.5, Review. |

May 10 | Sections 1.2-1.6, 2.1-2.5, Midterm. |

May 12 | Section 5.1, Eigenvalues and Eigenvectors. Notes. |

May 15 | Section 5.2, Diagonalizability. |

May 17 | Section 5.2, Diagonalizability. |

May 19 | Section 5.2, Diagonalizability. |

May 22 | Section 6.1, Inner Products and Norms. |

May 24 | Section 6.1, 6.2, Inner Products and Norms; The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. |

May 26 | Section 6.2, The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. |

May 31 | Section 6.3, The Adjoint of a Linear Operator. |

Jun 2 | Section 6.4, Normal and Self-Adjoint Operators. Notes. |

Jun 5 | Section 6.4, The Complex Spectral Theorem. Notes. |

Jun 7 | Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4, The Real Spectral Theorem. Notes. |

Jun 9 | Sections 1.2-1.6, 2.1-2.5, 4.4, 5.1-5.2, 6.1-6.4, Review. |

Jun 15 | 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 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.

Apr 3 | Homework 0. Deadline: Apr 7. |

Apr 3 | Homework 1. Deadline: Apr 14. |

Apr 10 | Homework 2. Deadline: Apr 21. |

Apr 17 | Homework 3. Deadline: Apr 28. |

Apr 24 | Homework 4. Deadline: May 5. |

May 1 | Homework 5. Deadline: May 12. |

May 8 | Homework 6. Deadline: May 19. |

May 15 | Homework 7. Deadline: May 26. |

May 22 | Homework 8. Deadline: Jun 2. |

May 29 | Homework 9. Deadline: Jun 9. |

Jun 5 | Homework 10. Deadline: Jun 11. |

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 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.

Apr 4 | Worksheet 1. Deadline: Apr 6. |

Apr 11 | Worksheet 2. Deadline: Apr 13. |

Apr 18 | Worksheet 3. Deadline: Apr 20. |

Apr 25 | Worksheet 4. Deadline: Apr 27. |

May 2 | Worksheet 5. Deadline: May 4. |

May 9 | Worksheet 6. Deadline: May 11. |

May 16 | Worksheet 7. Deadline: May 18. |

May 23 | Worksheet 8. Deadline: May 25. |

May 30 | Worksheet 9. Deadline: Jun 1. |

Jun 6 | Worksheet 10. Deadline: Jun 8. |

Exams

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

May 10 | Midterm, GEOLOGY 6704 from 11:00 to 11:50 am. |

Jun 15 | Final Exam, TBD from 11:30 am to 2:30 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 peer learning facilitator from the
**Academic Advancement Program**is Dinc Ozeren. - The open office hours at the Student Math Center can be found
**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.