# Math H54 - Honors Linear Algebra and Differential Equations

InstructorPaul Vojta
LecturesMWF 10–11 am, 3 Evans
Course Control Number31392
Office883 Evans
E-Mailvojta@math.berkeley.edu
Office HoursM 11:30–12:30, WF 11–12
PrerequisiteMath 1B
Required Text Lay, Nagle, Saff, & Snider, Linear Algebra and Differential Equations (UC Berkeley custom edition). See the bookstore web site for exact details.
Syllabus A syllabus was distributed in class on the first day, and is also available on the web at http://math.berkeley.edu/~vojta/syll.pdf.
 10% First midterm Wednesday, October 5 20% Second midterm Wednesday, November 2 45% Final exam Monday, December 12, 2016, 8:00–11:00am 25% Component from discussion section

The component from the discussion section is left to the discretion of the section leader, but it is likely to be determined primarily by quizzes and homework assignments (if the GSI decides to use the latter).

Note the final exam date given above. Do not enroll in this course if you cannot take the exam at that date and time, whether because of a conflict, too many exams on that day, or any other reason.

HomeworkHomework will consist of weekly assignments, taken from the book. You will not be required to hand them in (unless your GSI decides otherwise). You should do them before your discussion section meets on Wednesday of the following week (so, for example, the homework for Week 1 should be done by August 31).

The topics for the course will be:

• Basic linear algebra
• Matrix arithmetic and determinants
• Vectors in R2 and R3
• Vector spaces and 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

Most of the course consists of a detailed study of linear algebra. Linear algebra is, at its root, the study of systems of linear equations (you've probably encountered those before, both in high-school algebra courses and when doing partial fractions in Math 1B). Central to this study (for Math 54 and beyond) is the practice of putting the coefficients of a system of linear equations in the form of a matrix, and then doing operations on that matrix to better understand the system of equations. Math 54 goes quite a bit further, though, and looks at other things you can do with matrices. For example, recommendation systems (“People who bought this item also bought...”) use matrices in their computations (but in ways that go beyond what we'll do in Math 54).

The last (approximately) six weeks of the course involve differential equations. You should already be somewhat familiar with differential equations from Math 1B, where second-order linear differential equations with constant coefficients were studied. Math 54 goes a bit further, by studying higher-order linear differential equations, as well as first-order linear differential equations in more than one variable, and partial differential equations. The latter involve vibrations of fixed strings (as on a guitar or violin), vibrations of air in an organ pipe, etc. In other words, in Math 54 we study more kinds of differential equations than in Math 1B, and see how they relate to physical phenomena in the world around us.

A detailed syllabus will be passed out during the first week of class, and will also be posted on the web at around that time.

Honors Course

This course is aimed at students with a strong ability and interest in mathematics.

I will follow the curriculum for Math H54, but will try to provide greater rigor (real proofs), greater insight, and more interesting exercises. I don't expect the grading scale to be either higher or lower than for regular Math 54, but you will have to do more thinking to get a good grade; I hope that you will enjoy this. If you start H54 but find in a few weeks that it is not the course for you, it should be possible to transfer to regular Math 54 and not be at a disadvantage.

For more information on honors courses in the math department, see the department's web page on honors courses.

## Homework

Problem sets should be done before the discussion section on Wednesday of the following week, so that you can be prepared for that class. Answers to even-numbered exercises will be available on bCourses in pdf format later that day.

Homework assignments are the most important part of the course. Mathematics is not a spectator sport, and working on homework exercises is the best way to check that you really understand the material. In addition, many quiz questions will be closely based on homework exercises.

## Exams (Generally)

Policies for exams are as follows.

• No calculators or other portable electronic devices are allowed
• Exams are closed book. No notes or "cheat-sheets" are allowed.
• Do not bring bluebooks. Exams will have enough space for an answer to be written on the exam itself, and scrap paper will be available if you need it.

The two midterms will be given during the normal class hours (10–11 AM), and will be in our normal classroom (3 Evans).

Generally, about a week before each exam, a sample exam will be distributed in class and posted on bCourses. This will usually be an exam from an earlier Math 54 class that I've taught (I've never taught H54 before, so I have no previous H54 exams, but I'll try to modify the exams to account for the difference between regular Math 54 and Honors Math 54). Sample exams should be used to get a general idea of the likely length of an exam and the general nature of questions to be asked (e.g., the balance between computational and more theoretical questions). However, one should not (for example) note that a sample exam contains questions on material from Sections 1.5, 2.1, 2.7, 3.1, 3.4, etc., and expect to see questions from those sections on the actual exam.

Exams are cumulative, so the second midterm may have questions from material prior to the first midterm. Of course, the final exam will cover the whole course, but will have increased emphasis on the material not covered on the midterms.

Here is a link "How to lose marks on math exams" (by a former GSI Andrew Critch).

And finally, a word about regrades: Grade calculation errors are welcome for discussion or review. Whether this solution should be worth 4 or 5 points is not.

## First Midterm

The first midterm was given on Wednesday, October 5, at the normal class time (10:10–11:00 AM). It covered all material in the lectures and textbook in Chapters 1–4 (except for sections not listed on the syllabus).

A mock-up of the first page of the exam is available (the number of problems and points per problem were different).

A sample exam was distributed in class on Wednesday, September 28, and is also available on bCourses. Solutions for the sample midterm are also available on bCourses.

Solutions for the midterm itself are also available on bCourses.

The rough curve for the midterm is as follows:

 A 47–50 B 37–46 C 30–36 D 23–29

The median on the exam was 47, the mean was 41.4, and the standard deviation was 10.5. The median is more important than the mean. Also the distinction between A and B is not very significant for this exam.

Keep in mind that these letter grades are estimates only -- only the numbers are used to compute the final grade. Although it is tempting to try to predict the course grade by assigning points to letter grades and forming a weighted average, this method tends to predict grades that are more moderate (closer to a B or C) than the actual course grade ends up being.

## Second Midterm

The second midterm was given on Wednesday, November 2, at the normal class time (10:10–11:00 AM). It covered all material in the lectures and textbook in Chapters 1–6 and Section 7.1 (except for sections not listed on the syllabus). Most of the emphasis on the exam was on material that was not on the first midterm (Chapters 1–4).

A sample exam was distributed in class on Wednesday, October 26, and is available on bCourses. Solutions for the sample midterm, and to the midterm itself, are now available on bCourses.

The rough curve for the midterm is as follows:

 A 87–100 B 65–86 C 48–64 D 35–47

The median on the exam was 86, the mean was 76.9, and the standard deviation was 18.0. The median is more important than the mean.

Here is a list of parts of sections that were skipped.

SectionDescription
3.3Subsection "Application to engineering" (page 166)
5.1Subsection "Eigenvectors and difference equations" (page 241)
5.2Subsection "Application to dynamical systems" (pages 248–249)
7.1Subsection "Spectral Decomposition" (page 344)

## RRR Week

We will continue to meet in 3 Evans during RRR week at the usual class time. The time will be spent on reviewing the course material. (This should be regarded as a supplement, not as a replacement, for your own efforts to review the material.)

Prof. Vojta's office hours will continue at the usual times throughout RRR week.

## Final Exam

The final exam will be held on Monday, December 12, from 8:00 to 11:00 am. It will be held in our usual classroom (3 Evans).

About a half to two-thirds of the exam will be on material not covered on the midterms (i.e., differential equations).

Some formulas will be provided on the exam. They will be the same formulas as were given on the sample final exam. Note: Only the formulas are given. You will still need to know how to use them.

Here is a list of parts of sections that were skipped.

SectionDescription
3.3Subsection "Application to engineering" (page 166)
5.1Subsection "Eigenvectors and difference equations" (page 241)
5.2Subsection "Application to dynamical systems" (pages 248–249)
7.1Subsection "Spectral Decomposition" (page 344)
9.8We only did pages 551–553, plus versions of Example 1 when it is clear how to write A as a sum of a diagonalizable matrix and a nilpotent matrix
10.3 Theorem 3 (Uniform convergence of Fourier series, page 590) and Theorem 5 (Integration of Fourier series, page 591)
10.5Heat flow in higher dimensions (Example 4, pages 607–609) and Theorem 7 (Uniqueness of solution, page 610)
10.6Vibrating membrane (pages 615–616), traveling waves (d'Alembert's solution, pages 616–619), and existence and uniqueness (pages 620–622)
10.7Skipped completely

A practice final exam was handed out in class on Monday, December 5 (RRR week), and is also available on bCourses. Solutions to it will be posted on Thursday.

See also the review sheets for each of the exams (lists of definitions, etc.).

## Old Exams

Sample exams will be distributed in class shortly before each exam.

The Math Department maintains an archive of old exams (usually without answers). Here is the link for Math 54 There appears to be no file for H54.

Last updated 5 December 2016