MATH54

UC Berkeley Math 54, Fall 2022

This is a repository for the course Math 54: Linear Algebra & Differential Equations in Fall 2022.

This repository contains most of the information you need for this course (lecture notes, assignments). Course sensitive information (announcements, exam, grade distribution etc) will be posted on bCourses (CalNet ID required).

If you are taking a makeup final exam from a previous semester or need to complete the Math 49 assignment, please see the end of the webpage for instructions on how to proceed with the final exam.

General information

Please read the Course Policy VERY CAREFULLY

For administrative questions (not covered in the course policy): please email the Lead GSI first: Jiahao Yao jiahaoyao@berkeley.edu

Instructor: Lin Lin

Lecture: MonWedFri 09:10AM - 10:00AM Dwinelle 155

Office Hours: Mon 10:10AM-11:00AM, Wed 3:10PM-4PM. Evans 817

Important dates

Midterm 1: 9/23 Fri. 9:10AM-10AM in class.

Midterm 2: 10/28 Fri. 9:10AM-10AM in class.

Final: 12/15 Thu 7-10 pm PST. Location TBD.

Textbook: Linear algebra and differential equations, Second Custom Edition for UC Berkeley. ISBN: 9780137114030

textbook_1

The new textbook contains chapters from the 5th and 6th (new to this custom version) editions of Lay, Lay, & McDonald’s Linear Algebra and Its Applications and chapters from the 9th edition of Nagle, Saff and Snider’s Fundamentals of Differential Equations. The original custom edition only included chapters from the 5th edition of Lay, Lay, & McDonald’s Linear Algebra and Its Applications, and chapters from the 9th edition of Nagle, Saff and Snider’s Fundamentals of Differential Equations.

Previous versions of the custom edition, or the separate textbooks as above may work (I do not have earlier versions of the textbooks with me so I cannot explain the detailed differences), and it is your responsibility to make sure that you are doing the correct problem sets for your homework.

GSI office hours:

Lewis Pan, Monday/Wednesday 12-1 PM, Evans 1070

Katalin Berlow, Tuesday/Thursday 6:30-7:30 PM, Evans 762

Ovidiu-Neculai Avadanei, Monday and Wednesday 1-2 PM, Evans 869

Jiasu Wang, Tuesday 2-4pm, Evans 739.

Yulong Dong, Tuesday 11:50-12:20pm and 3:30-5pm, Evans 787

Chan Bae, Tuesday/Thursday 7-8 PM, Evans 1066

Robert Schütz, Wed 11-1 in Evans 961.

Rose Lopez, Mondays 1-3, Evans 1061

Jiahao Yao, Fri, 4 - 5 pm, Evans 935

GSI emails:

Jiahao Yao jiahaoyao@berkeley.edu,

Lewis Pan yllpan@berkeley.edu,

Jiasu Wang jiasu@berkeley.edu,

Ovidiu-Neculai Avadanei ovidiu_avadanei@berkeley.edu,

Yulong Dong dongyl@berkeley.edu,

Katalin Berlow katalin@berkeley.edu,

Chan Bae c_h_bae@berkeley.edu,

Robert Schutz rschuetz@berkeley.edu,

Rose Lopez roselopez@berkeley.edu

Piazza page: General questions about the course and its content, which might be of interest to other students, can be asked on the piazza page.

Additional resources:

Part 1. Linear algebra, first half.

General information of the class (pdf)

Lecture 8/24 Wed

Systems of linear equations, 1.1

Homework due 8/30 Tue: 1.1: 1, 7, 9, 11, 20, 21, 26, 27, 28, 29, 32, 33, 34, 38

No Quiz on 8/25 Thu

Lecture note 1 (pdf)

Lecture 8/26 Fri

Row reduction 1.2.

Homework due 8/30 Tue: 1.2: 3, 7, 11, 15, 19, 23, 26, 27, 29, 30, 34, 45

Lecture note 2 (pdf)

Lecture 8/29 Mon

Vectors and matrices. 1.3, 1.4.

Homework due 9/6 Tue:

1.3: 5, 11, 14, 18, 25, 26, 27, 29, 30, 32, 33, 37, 38

1.4: 3, 10, 13, 15, 17, 18, 23, 24, 27, 28, 31, 32, 34, 36, 45

Lecture note 3 (pdf)

Lecture 8/31 Wed

Solution sets of linear systems. 1.5

Linear independence. 1.7

Homework due 9/6 Tue: 1.5: 3, 8, 17, 21, 24, 27, 29, 31, 32, 33, 34, 36, 37, 47, 50

Quiz on 9/1 Thu covers: 1.1-1.2

Lecture note 4 (pdf)

Lecture 9/2 Fri

Linear transformations. 1.8, 1.9

Homework due 9/6 Tue: 1.7: 1,8,9,11,14,16,21,22,24,25,26,27,28,32,35,38,39,41,42,46

Lecture note 5 (pdf)

No Lecture 9/5 Mon (Labor Day)

The optional homework assignments are not required to be submitted to your GSI. These optional assignments can help consolidate your understanding of the course material.

Lecture 9/7 Wed

Matrix operations. 2.1.

Homework due 9/13 Tue:

1.8: 9,22,24,33,37

1.9: 4,15,24,26,43

2.1: 12,18,19,26,29,32,42

Optional:

1.8: 1,3,17,19,20,21,25,27,28,30,34,36,38

1.9: 5,6,10,11,14,23,27,29,32,34,40,

2.1: 2,6,9,11,15,16,17,22,23,24,31,35,37,38,39

Quiz on 9/8 Thu covers: 1.3-1.7

Lecture note 6 (pdf)

Lecture 9/9 Fri

Matrix inverse and invertible matrices 2.2, 2.3

Homework due 9/13 Tue:

2.2: 10,12,18,27,29

2.3: 4,11,19,21,28

Optional:

2.2: 2,13,16,17,20,30,37,43,44

2.3: 12,13,15,30,35,45

Lecture note 7 (pdf)

Lecture 9/12 Mon

Subspaces, 2.8, Dimension and rank, 2.9.

Homework due 9/20 Tue:

2.8: 8,17,25,28,36

2.9: 14,17,22,26,35

Optional:

2.8: 5,7,9,15,21,22,23,26,31,41,43

2.9: 3,5,7,9,11,16,19,23,24,32,33

Lecture note 8 (pdf)

Lecture 9/14 Wed

Vector space, 4.1.

Homework due 9/20 Tue:

4.1: 9,13,20,23,32,41

Optional:

4.1: 1,3,14,19,21,24,27,31,32,37,40

Quiz on 9/15 Thu covers: 1.8-2.3

Lecture note 9(pdf)

Optional:

Now we start to see something cool about linear algebra in real life:

Read this article on Word2Vec if you want to understand how computers can figure out the meaning of equations like

“King - Man + Woman = Queen”

using linear algebra. If you want to know even more about this, watch this video.

Lecture 9/16 Fri

Null space, column spaces and linear transformations, 4.2

Homework due 9/20 Tue:

4.2: 15,37,42,43,45

Optional:

4.2: 2,3,7,12,25,26,31,32,33,34

Lecture note 10(pdf)

Lecture 9/19 Mon

Bases, coordinates 4.3, 4.4

Homework due 9/27 Tue:

4.3: 8,11,19,23,30,36

4.4: 13,16,24,35,36

Optional:

4.3: 3,4,13,15,21,22,23,24,32,34,35

4.4: 3,6,11,15,17,22,26,28,30,31

Lecture note 11(pdf)

Lecture 9/21 Wed

Dimension, rank 4.5

Homework due 9/27 Tue:

4.5: 6,12,18,20,25,27,45,52

Optional:

4.5: 3,11,19,21,22,23,24,29,32,34,39,43,44

No Quiz on 9/22 Thu

Lecture note 12(pdf)

Midterm 1: 9/23 Fri

Midterm #1 in class, covers: 1.1-2.9 (Materials in Chapter 4 are not included in Midterm 1 but will appear in Midterm 2)

List of topics in Midterm 1 (pdf) : this is the Study Sheet written by Prof. Nikhil Srivastava. I cannot do a better job than him in terms of summarizing and clarifying the key definitions, theorems, algorithms, and types of problems that you should know. Please read this carefully. (Chapter 3 is not covered in Midterm 1)

More resources:

Prof. Alexander Paulin has a nice collection of practice exams and solutions .

Practice Midterm 1#1 and solutions (Prof. Nikhil Srivastava)

Practice Midterm 1#2 and solutions (Prof. Nikhil Srivastava)

Practice Midterm 1#3 and solutions (Prof. Nikhil Srivastava)

Part 2. Linear algebra, second half.

Lecture 9/26 Mon

Change of basis, 4.6. Matrix representation of linear transformation (part of 5.4)

Homework due 10/4 Tue:

4.6: 5,10,14,16

5.4: 3,5,6

Lecture note 13(pdf)

Optional: 4.6: 2,3,11,12,13,17

In the class we watched part of a video describing the geometric idea behind determinants. You should also watch this video explaining the idea behind negative area in 2D (from 3:50) and in 3D (from 6:50).

Optional:

If you have not watched the videos by 3Blue1Brown before, I also highly recommend his videos on

Linear combination, Abstract vector spaces, and many more!

Lecture 9/28 Wed

Determinants 3.1, 3.2

Homework due 10/4 Tue:

3.1: 4,11,22,33,42,

3.2: 5,16,22,28,32,33

Quiz on 9/29 Thu covers: 2.8-4.5

Lecture note 14(pdf)

Optional:

3.1: 1,5,9,10,31,37,40,41

3.2: 10,17,18,21,37,38,43,48

Watch this video (only the part 6:00 to 12:00) on determinantal point process. You can also type “jaguar” to Google Image and see the results.

Lecture 9/30 Fri

More about determinants, 3.3

Homework due 10/4 Tue:

3.3: 1,7,19,27,30,35,37

Lecture note 15(pdf)

Optional:

3.3: 7,18,23,36

Watch this video for the geometric explanation of Cramer’s rule (starting from 5:30. It is better than the book!)

Lecture 10/3 Mon

Eigenvectors and eigenvalues. 5.1

Homework due 10/11 Tue:

5.1: 10,18,21,24,35,37

Lecture note 16(pdf)

Watch the remaining part of this video for eigenvalues and eigenvectors.

Optional:

Review how Tony Stark saved the universe by computing the eigenvalue of a Mobius strip.

Watch this video to see how Google’s PageRank works and how it is related to an eigenvalue problem.

If you want to know more, try to read this paper (and be amazed that once upon a time, Google was only worth 25 billion dollars!)

Lecture 10/5 Wed

Characteristic equation 5.2

Homework due 10/11 Tue:

5.2: 9,16,21,26,29,32

Lecture note 17(pdf)

Quiz on 10/6 Thu covers: 4.6, Part of 5.4, 3.1-3.3

Lecture 10/7 Fri

complex eigenvalue 5.5

Homework due 10/11 Tue:

5.5: 2,7,23,27

Lecture note 18(pdf)

Lecture 10/10 Mon

Diagonalizability 5.3

Homework due 10/18 Tue:

5.3: 3,19,26,28,31,34

Lecture note 19(pdf)

Lecture 10/12 Wed

Linear transformation, similarity transformation 5.4

Homework due 10/18 Tue:

5.4: 9,15,16,20,21,27

Quiz on 10/13 Thu covers: 5.1-5.3, 5.5

Lecture note 20(pdf)

Lecture 10/14 Fri

Inner product 6.1

Homework due 10/18 Tue:

6.1: 14, 22, 25, 28, 32, 36, 38

Lecture note 21(pdf)

Lecture 10/17 Mon

Orthogonal sets 6.2

Homework due 10/25 Tue:

6.2: 12, 22, 26, 32, 37, 42

Lecture note 22(pdf)

Lecture 10/19 Wed

Orthogonal projections 6.3, Least-squares problems 6.5

Homework due 10/25 Tue:

6.3: 15, 17, 22, 25, 32

6.5: 2, 9, 18, 21, 27, 29

Quiz on 10/20 Thu covers: 5.4, 6.1

Lecture note 23(pdf)

Lecture 10/21 Fri

Gram-Schmidt process 6.4

Homework due 10/25 Tue:

6.4: 9, 17, 20, 26

Use least squares to study the early phase of the COVID-19 pandemic in US (ipynb notebook)

Lecture note 24(pdf)

Lecture 10/24 Mon

General inner product spaces 6.7

Homework due 11/1 Tue:

6.7: 3, 9, 12, 13, 17, 31

Lecture note 25(pdf)

Lecture 10/26 Wed

General (complex) inner product spaces 6.7

Homework due 11/1 Tue: None

Lecture note 26(pdf)

No Quiz on 10/27 Thu:

Midterm 2: 10/28 Fri

Midterm #2 in class, covers: 3.1-6.5

List of topics in Midterm 2 (pdf) : this is the Study Sheet written by Prof. Nikhil Srivastava. I cannot do a better job than him in terms of summarizing and clarifying the key definitions, theorems, algorithms, and types of problems that you should know. Please use this as a reference and read this carefully.

More resources:

Prof. Alexander Paulin has a nice collection of practice exams and solutions .

From Prof. Nikhil Srivastava:

Practice Midterm 2#1 and solutions

Practice Midterm 2#2 and solutions

Practice Midterm 2#3 and solutions

Lecture 10/31 Mon

Diagonalization of symmetric matrices 7.1

Homework due 11/8 Tue:

7.1: 21, 24, 30, 36, 38

Lecture note 27(pdf)

Lecture 11/2 Wed

Singular value decomposition 7.4

Homework due 11/8 Tue:

7.4: 11, 13, 17, 20, 25

Singular value decomposition based compression of an image (juia notebook)

Lecture note 28(pdf)

Quiz on 11/3 Thu covers: 6.2-6.7

Part 3. Differential equations.

From now on all chapters numbers refer to NS&S

Lecture 11/4 Fri

Homogeneous linear equations 4.2

Homework due 11/8 Tue:

4.2: 5, 16, 26, 35, 42

Lecture note 29(pdf)

Spring simulator

Lecture 11/7 Mon

Homogeneous linear equations 4.3

Homework due 11/15 Tue:

4.3: 11, 28, 31, 35

Lecture note 30(pdf)

Lecture 11/9 Wed

Inhomogeneous linear equations 4.4, 4.5

Homework due 11/15 Tue:

4.4: 1, 4

4.5: 17, 32

Quiz on 11/10 Thu covers: Lay 7.1, 7.4; NS&S 4.2

Lecture note 31(pdf)

No Lecture 11/11 Fri (Veterans Day)

Lecture 11/14 Mon

Inhomogeneous linear equations 4.4, 4.5

Homework due 11/22 Tue: None.

Lecture note 32(pdf)

Lecture 11/16 Wed

Inhomogeneous linear equations 4.4, 4.5

Homework due 11/22 Tue:

4.4: 13, 24

4.5: 2, 11, 33

Lecture note 33(pdf)

Quiz on 11/17 Thu covers: NS&S 4.3-4.4

Lecture 11/18 Fri

Linear systems in normal form 9.4

Homogeneous linear systems 9.5, 9.6

Homework due 11/22 Tue:

9.5: 12, 18

9.6: 5, 13

Lecture note 34(pdf)

Lecture 11/21 Mon

Review session, Q&A (Zoom only)

https://berkeley.zoom.us/j/92646132059

Lecture note 35(pdf)

No Lecture 11/23 Wed (Thanksgiving)

No Lecture 11/25 Fri (Thanksgiving)

Lecture 11/28 Mon

Review session, Q&A (Zoom only)

https://berkeley.zoom.us/j/92646132059

Lecture 11/30 Wed

Review session, Q&A (Zoom only)

https://berkeley.zoom.us/j/92646132059

Lecture 12/2 Fri (Formal Classes End)

Review session, Q&A (Zoom only)

https://berkeley.zoom.us/j/92646132059

No Lecture RRR week 12/5-12/9

Final exam 12/15 Thu 7pm-10 pm PT

Prof. Alexander Paulin has a nice collection of practice exams and solutions .

Plan for final exam: waiting for campus instruction

There is no make-up final exam

Room (7pm-10 pm PT): Pimentel Hall, Room 1

Room for DSP students (4pm-10pm PT): Moffitt,Room 101

If you are taking a makeup final exam (from Professor Paulin’s class last semester), please 1) join gradescope with the course entry code Entry V5W3WB 2) directly come to Pimentel Hall, Room 1 to take the final exam. Please do not email me asking for the final score. Your final scores will be directly sent to Professor Paulin at the end of the semester.

If you need to complete the Math 49 assignment, please 1) join gradescope with the course entry code Entry V5W3WB 2) directly come to Pimentel Hall, Room 1 to take the final exam. If the Math 49 assignment says that you only need to finish the differential equation part, then you only need to work on the questions in the final exam related to differential equations. Please do not email me asking for the final score. At the end of the semester, you can contact the Lead GSI Jiahao Yao jiahaoyao@berkeley.edu to finish your Math 49 requirement.