# Math 54, Summer 2016

This is the homepage for the Summer 2016 Math 54 (Lecture 3) and its discussion section.

**Instructor:** Kyle Miller

**E-mail:** kmill at math.berkeley.edu

**Meeting times:**

Lecture: MTWHF 12-1 pm in 109 Morgan

Discussion: MTWHF 1-2 pm in 109 Morgan

**Office hours:** Wed/Thu 3-5 pm in 1066 Evans, or by appointment

**bCourses:**
https://bcourses.berkeley.edu/courses/1451943

**Piazza:**
https://piazza.com/berkeley/summer2016/math54/home

The math department publishes a rough course overview. There is also a collection of some past exams.

## 1. Syllabus

**Prerequisites:** Math 1A-1B or equivalent. Students who have
taken such courses at another institution may not have studied
differential equations and should learn the relevant material
(Stewart *Single Variable Calculus* chapters 9 and 17) on
their own by the time we get to the differential equations portion
of Math 54.

**Official description:** Basic linear algebra; matrix arithmetic and
determinants. Vector spaces; inner product spaces. Eigenvalues and
eigenvectors; linear transformations. Homogeneous ordinary
differential equations; first-order differential equations with
constant coefficients. Fourier series and partial differential
equations.

**Textbook:** Lay-Nagle-Saff-Snider, *Linear Algebra &
Differential Equations*. A specially priced UC Berkeley paperback
edition is available. This is a combination of

- Lay,
*Linear Algebra and Its Applications*(4th ed.); and - Nagle, Saff, and Snider,
*Fundamentals of Differential Equations*(8th ed.).

### 1.1. Homework, quizzes, and exams

Homework will be due once per week, at the end of discussion Friday. Late homework will be given points according to the formula ${2}^{-n}s$, rounded down, where $s$ is the score the homework would have received when not late, and $n$ is the number of days late.

We will have fifteen-minute quizzes every Friday at the beginning of discussion. There will be no make-up quizzes.

A one-hour midterm will be given July 22nd during lecture.

A two-hour final will be given August 12 during lecture, through discussion.

The lowest quiz score and the lowest homework score will be dropped.

### 1.2. Grading policy

The final grade will be determined according to the following: 15% homeworks, 20% quizzes, 25% midterm, 40% final exam.

### 1.3. Special accommodations

All students requesting special accommodations need to be registered with the Disabled Student Services and need to provide a letter indicating the necessary accommodations. Because it is very difficult to arrange accommodations at the last minute, all requests must be received at least 8 days prior to the exams.

### 1.4. Piazza

This term we will be using Piazza for class discussion. I encourage you to ask questions on Piazza rather than via e-mails for our collective benefit. Find our class page at https://piazza.com/berkeley/summer2016/math54/home.

## 2. Schedule

*Note:* Answers to odd-number exercises for Lay begin on
page 363. For the second textbook, page 657.

This schedule is subject to change. Homework, once posted, will remain constant.

### 2.1. Week 1 (Jun 20–24)

Reading: sections 1.1–1.5, 1.7–1.9 (Lay)

Homework 1: (Lay) due June 24

1.1: 1, 3, 9, 11, 17, 18, 23

1.2: 1, 3, 4, 13, 15, 19, 21

1.3: 1, 5, 7, 12, 14, 25

1.4: 1, 2, 5, 11, 23, 29, 30

1.5: 1, 5, 7, 10, 23, 33

1.7: 1, 3, 6, 11, 15, 21

### 2.2. Week 2 (Jun 27–Jul 1)

Reading: sections 2.1–2.3, 3.1–3.3, 4.1 (Lay)

Homework 2: (Lay) due July 1

1.8: 1, 3, 5, 7, 9, 32, 24

1.9: 1, 3, 5, 7, 17, 23

2.1: 1, 5, 7, 8, 9, 17

2.2: 1, 4, 5, 8, 9, 31

2.3: 1, 8, 15, 16, 20, 33

3.1: 1, 2, 6, 9, 25, 27, 29

3.2: 5, 11, 22, 27, 37, 39

3.3: 1, 8, 12, 19, 29

### 2.3. Week 3 (Jul 5–8)

*Note: July 4 is a holiday.*

Reading: sections 4.2–4.7 (Lay)

Homework 3: (Lay) due July 8

4.1: 1, 3, 6, 11, 13, 21, 31

4.2: 1, 4, 8, 12, 24, 31, (35, 36)

4.3: 11, 13, 19, 23, 24, 31, 32

4.4: 1, 9, 12, 13, 15

4.5: 1, 10, 11, 13, 14, 29

*Problems in parentheses are optional and ungraded. Please read them, at
least.*

### 2.4. Week 4 (Jul 11–15)

Reading: sections 5.1–5.4, 6.1 (Lay)

Homework 4: (Lay) due July 15

4.6: 1, 5, 8, 16, 28

4.7: 7

5.1: 1, 3, 11, 13, 16, 26

5.2: 1, 2, 9, 10, 18, 20

5.3: 1, 7, 10, 17, 18, 27

5.4: 10, 13, 14, 22, 25, 26

(5.5: 7, 17, 21, 22, 23)

6.1: 1, 2, 15, 16, 27, 28

*Problems in parentheses are optional and ungraded. They are
fairly important, so it is worth at least reading them.*

### 2.5. Week 5 (Jul 18–22)

Reading: sections 6.2–6.5, 6.7, 7.1 (Lay)

Homework 5: (Lay) due July 25 (Monday)

6.2: 1, 2, 8, 11, 19

6.3: 1, 3, 12, 18

6.4: 1, 2, 13, 18

6.5: 1, 10, 19, 20, 21

6.7: 3, 13, 25 (clarification: Gram-Schmidt $1$, $t$ ,${t}^{2}$)

7.1: 13, 14, 23

Midterm: July 22 in lecture.

### 2.6. Week 6 (Jul 25–29)

Reading: sections 4.1–4.6, 6.1 (NSS)

Homework 6: (**NSS**) due July 29

4.1: 4, 7

4.2: 1, 2, 13, 14, 22, 29, 37

4.3: 2, 3, 9, 21, 38

4.4: 9, 12, 14, 18, 28, 30, 34

4.5: 1, 4, 7, 9, 18, 23

4.6: 3, 4, 20

### 2.7. Week 7 (Aug 1–5)

Reading: sections 6.2, 9.4–9.8 (NSS)

Homework 7: (**NSS**) due August 5

6.1: 7, 9, 23

6.2: 1, 6, 15, 19

9.1: 1, 3, 7, 11

9.4: 14, 28, 30

9.5: 2, 9, 12, 31, 32, 50

9.6: 1, 13

9.7: 1, 2, 8, 21(a), 24

9.8: 7, 8, 12, 17, 22

### 2.8. Week 8 (Aug 8–12)

Reading: sections 10.1–10.7 (NSS)

Final: August 12 in lecture.

## 3. Lecture notes

- June 20: linear systems.
- June 21: matrices. Discussion.
- June 22: vectors. Discussion.
- June 23: existence, homogeneous systems, independence. Discussion. Row reducer.
- June 24: independence, linear transformations. Discussion. Cat transformation.
- June 27: one-to-one, onto, matrix multiplication. Discussion.
- June 28: matrix multiplication, transposes, inverses. Discussion.
- June 29: inverse matrix theorem, determinants. Discussion.
- June 30: determinants, Cramer’s rule. Discussion. Interactive cofactor expansion
- July 1: vector spaces. Discussion.
- July 5: subspaces, nullspace, column space, kernel, image. Discussion.
- July 6: independence, bases. Discussion.
- July 7: coordinates, dimension. Discussion. Bases and coordinates in ${\mathbb{R}}^{2}$
- July 8: rank, change of basis.
- July 11: eigenvectors, eigenvalues, characteristic equation. Discussion.
- July 12: similarity, diagonalization. Cat transformation: eigenvector edition.
- July 13: diagonalization, matrix of a transformation, complex numbers. Discussion.
- July 14: dot product, inner product, orthogonality, orthogonal complements. Discussion.
- July 18: orthogonal matrices, projection.
- July 19: Gram-Schmidt process, QR factorization.
- July 20: Least-squares, inner products.
- July 21: Spectral theorem, orthogonal diagonalization. Midterm exam schemata
- July 25: Differential equations, second-order homogeneous linear differential equations.
- July 26: Existence and uniqueness, independence, Wronskian.
- July 27: Periodic solutions, method of undetermined coefficients.
- July 28: Superposition, non-homogenous second-order linear differential equations, variation of parameters.
- July 29: Higher-order linear differential equations.
- August 1: Higher-order linear differential equations with constant coefficients.
- August 2: Systems of linear differential equations.
- August 3: Homogeneous systems of linear differential equations.
- August 4: Complex eigenvalues, nonhomogeneous systems of linear differential equations.
- August 5: Matrix exponential.
- August 8: Partial differential
equations, the heat equation.
^{[1]}Heat equation. - August 9: The wave equation.
- August 10: Fourier series.

## 4. Programs

The following are some programs to illustrate some computations and concepts from the course. These are linked to in the Lecture notes section.- Row reducer
- Cat transformation
- Interactive cofactor expansion
- Bases and coordinates in ${\mathbb{R}}^{2}$
- Heat equation
- Wave equation

## 5. Worksheets

The department publishes worksheets on the Lower Division Course Outlines page, Math 54 Worksheet (pdf).

## 6. How to study

**Step 1.** Read the book, go to lecture, section, and office
hours, solve examples. Check your understanding.

**Step 2.** Solve homework problems. If you get stuck, you did
not complete Step 1, so go back and figure out exactly what you
still need to understand.

Mathematics does not end with words written on a page, rather an answer is an artifact or byproduct of understanding. Furthermore, you are trying to convice the reader that you are correct: thus revise! It is your responsibility to make your answers clear. In fact, if on a quiz or exam the grader does not notice you are “right,” this is your fault, and you will lose points for it.