Table of Contents | |

1. Description | |

2. Class Notes | |

3. Schedule | |

4. Homework | |

5. Discussion | |

6. Exams | |

7. Resources |

UNIVERSITY OF CALIFORNIA, LOS ANGELES - WINTER 2022

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)

Table of Contents | |

1. Description | |

2. Class Notes | |

3. Schedule | |

4. Homework | |

5. Discussion | |

6. Exams | |

7. Resources |

Description

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 BUNCHE, room 1209B, from 11:00 to 11:50 am. The online classroom is Zoom Meeting 945 8336 4011.**

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.

- Vocabulary in Linear Algebra.
- Midterm 1 Review on 23 January 2022.
- Midterm 1 Review on 27 January 2022.
- Midterm 2 Review on 12 February 2022.
- Midterm 2 Review on 16 February 2022.
- Change of coordinates by
**Jas Singh**. - Diagonalization and the Fibonacci sequence by
**Jacob Swenberg**. - Essence of linear algebra by
**3Blue1Brown**. - Isometries of the plane and linear algebra by
**Keith Conrad**.

Tentative Schedule

Jan 3 | Sections 1.1, 1.2, Introduction to Linear Systems, Matrices, Vectors, and Gauss-Jordan Elimination. Notes. Office hours. |

Jan 5 | Section 1.3, On the Solutions of Linear Systems; Matrix Algebra. Notes. |

Jan 7 | Section 2.1, Introduction to Linear Transformations and Their Inverses. Notes. |

Jan 10 | Section 2.2, Linear Transformations in Geometry. Notes. Office hours. |

Jan 12 | Section 2.3, Matrix Products. Notes. |

Jan 14 | Section 2.4, The Inverse of a Linear Transformation. Notes. |

Jan 19 | Section 3.1, Image and Kernel of a Linear Transformation. Notes. Office hours. |

Jan 21 | Section 3.2, Subspaces of ℝ ^{n}; Bases and Linear Independence. Notes. |

Jan 24 | Section 3.3, The Dimension of a Subspace of ℝ ^{n}. Notes. |

Jan 26 | Section 3.4, Coordinates. Notes. |

Jan 28 | Sections 1.1-1.3, 2.1-2.4, 3.1-3.4, Midterm 1. |

Jan 31 | Section 5.1, Orthogonal Projections and Orthonormal Bases. Notes. Office hours. |

Feb 2 | Section 5.1, Orthogonal Projections and Orthonormal Bases (continued). Notes. |

Feb 4 | Section 5.2, Gram-Schmidt Process and QR Factorization. Notes. |

Feb 7 | Section 5.3, Orthogonal Transformations and Orthogonal Matrices. Notes. |

Feb 9 | Section 5.4, Least squares. Notes. Office hours. |

Feb 11 | Section 6.1, Introduction to Determinants. Notes. |

Feb 14 | Section 6.2, Properties of the Determinant. Notes. |

Feb 16 | Section 6.3, Geometrical Interpretations of the Determinant; Cramer's Rule. Notes. Office hours. |

Feb 18 | Sections 5.1-5.4, 6.1-6.3, Midterm 2. |

Feb 23 | Section 7.1, Diagonalization. Notes. |

Feb 25 | Section 7.2, Finding the Eigenvalues of a Matrix. Notes. |

Feb 28 | Section 7.3, Finding the Eigenvectors of a Matrix. Notes. |

Mar 2 | Section 7.5, Complex Eigenvalues. Notes. |

Mar 4 | Section 8.1, Symmetric Matrices. Notes. |

Mar 7 | Section 8.2, Quadratic Forms. Notes. |

Mar 9 | Section 8.3, Singular Values. Notes. Office hours. |

Mar 11 | Sections 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 12 | Sections 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. |

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, deadlines, and solutions will be posted here.

Jan 3 | Homework 0. Deadline: Jan 11. |

Jan 4 | Homework 1. Deadline: Jan 12. |

Jan 11 | Homework 2. Deadline: Jan 19. |

Jan 18 | Homework 3. Deadline: Jan 26. |

Jan 25 | Homework 4. Deadline: Feb 2. |

Feb 1 | Homework 5. Deadline: Feb 9. |

Feb 8 | Homework 6. Deadline: Feb 16. |

Feb 15 | Homework 7. Deadline: Feb 23. |

Feb 22 | Homework 8. Deadline: Mar 2. |

Mar 1 | Homework 9. Deadline: Mar 9. |

Mar 8 | Homework 10. Deadline: Mar 16. |

Discussion

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 4 | Worksheet 1, Solutions. |

Jan 11 | Worksheet 2, Solutions. |

Jan 18 | Worksheet 3, Solutions. |

Jan 25 | Worksheet 4, Solutions. |

Feb 1 | Worksheet 5, Solutions. |

Feb 8 | Worksheet 6, Solutions. |

Feb 15 | Worksheet 7, Solutions. |

Feb 22 | Worksheet 8, Solutions. |

Mar 1 | Worksheet 9, Solutions. |

Mar 8 | Worksheet 10, Solutions. |

Exams

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

Jan 28 | Midterm 1, Solutions, BUNCHE 1209B from 11:00 to 11:50 am. |

Feb 18 | Midterm 2, Solutions, BUNCHE 1209B from 11:00 to 11:50 am. |

Mar 12 | Final Exam, Solutions, TBD from 8:00 to 11:00 am. |

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 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.