(See also (currently useless) web page on Courseworks.)

Time: TTh 1:10-2:25
Place: Math 417
Textbook: Linear Algebra by Klaus Jänich. See note, below.
Office hours: Monday, 10:00-11:00 a.m., 12:00-1:00 p.m.
Teaching assistant: Evan Fink
Help room hours: Thursday, 3:00-6:00
Final exam: Tuesday Dec. 16, 1:10-4:00 p.m. in Math 417.

#### Announcements:

• 12/04: Solutions to all remaining homeworks, and second midterm, have been posted.
• 11/13: Syllabus revised.
• 10/30: Solutions to homework 7 posted.
• 10/29: Some review problems for the second midterm are now on Courseworks, in the class files section. (Also solutions to homeworks 5, 6, exam, and soon 7.)
• 10/15: Reading for Monday posted in syllabus section and on Courseworks
• 10/9: Problem set 5 updated, to reflect what we covered in class, and make problem 1 easier.
• 10/2: Solutions to homework 3, 4, review questions all on courseworks.
• 09/19: Note that office hours have changed.
• 09/16: There's an Undergraduate Math Society talk this Wednesday at 7 p.m. which you might find interesting, by Aaron Lauda.
• 09/16: The help room hours are here.
• 09/16: Solutions to HW2 now posted on Courseworks, in the "Class Files" section.
• 09/09: Solutions to HW1 now posted on Courseworks, in the "Class Files" section.
Syllabus Problem sets Policies Advice

## Description and goals.

At heart, linear algebra is about linear equations and linear transformations. Linear algebra's importance to both mathematics and its applications rivals -- and perhaps exceeds -- that of calculus. Unlike calculus, however, linear algebra becomes clearer in a somewhat more abstract setting -- that of vector spaces, linear transformations and inner products. This course will discuss abstract linear algebra. Two things will keep us anchored: a multitude of concrete examples, computations and applications; and proofs, to track what is true in general -- and why.

The main goals of the course are:

• To provide the essential tools from linear algebra needed in mathematics, science and engineering.
• To provide an introduction to writing and reading mathematics, suitable for taking more advanced courses or studying further mathematics on one's own.
• To begin to explore a deep and beautiful subject, and its relationships with other parts of mathematics and science.

## How is V2020 different from V2010?

This is the first time V2020 is being taught, so it is a bit of an experiment. While there will be a lot of overlap between "Linear Algebra" and "Honors Linear Algebra," there will also be several differences. In particular, V2020 will:

• Present somewhat more material than V2010. In particular, we will talk about Jordan Normal Form, the matrix exponential, and several versions of the Spectral Theorem -- and applications of these.
• Work in a more abstract setting -- that of vector spaces and linear maps -- than V2010. This requires a change in mind set at the beginning, which can be difficult. Ultimately, however, the ideas are clearer from a more abstract perspective.
• Involve writing proofs. We will work up to this as gently as possible -- but by the end of the semester students should be able to write clear mathematical arguments. (Students are *not* expected to have had experience writing proofs before taking the course.)

Consequently, the material in V2020 will be presented at a somewhat higher pace than in V2010. Although students in V2020 will learn to compute everything that students in V2010 do, there will be somewhat less practice computing. We will also talk about some applications of linear algebra -- probably including to linear differential equations, Markov processes, and perhaps statistics -- but may not spend as much time on applications as V2010.

You should strongly consider V2020 instead of V2010 if:

• You are planning to major in mathematics.
• You enjoy thinking about (abstract?) mathematical questions.
• You would like a sense of what more advanced mathematics courses are like.
• You are planning to take more advanced mathematics courses later.
• You want a deeper background in mathematics for your major or other studies.

You should not take V2020 if:

• You already took V2010 or V1207-1208.
• You do not like thinking about abstract mathematics (or strongly prefer numbers to variables, say).
• You are only taking linear algebra because of a degree requirement.

As noted above, the textbook for the course will probably be Klaus Jänich's Linear Algebra. If you are still not sure which course to take, you might get hold of a copy and read the second chapter, to see if you like it. Another, similar book is Linear Algebra Done Right by Sheldon Axler, and you could look at the first chapter of that to see if you like it. (The Axler book is available online through the Columbia
libraries -- search in Clio.)

## Policies.

### Grading

 Homework 25% Midterm 1 20% Midterm 2 20% Final exam 30% Class participation, bonus problems, office hour attendance 5%

The lowest homework score will be dropped.

### Homework

Problem sets are due on Tuesdays at the beginning of class, except as noted below. If you can't make it to class, put it in my mailbox before class in Math 509 (to the right of the door when you enter).

You're welcome to work on the homework together. However, you must write up your final answers by yourself. I consider writing them up together cheating.

You are also generally welcome to use any resources you like to solve the problems. However, any resource you use other than the textbook (Jänich) must be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including the help room and your classmates). Failure to cite sources constitutes academic misconduct.

Active reading check. Each day you are expected to read the sections listed in the syllabus before class. Note down at least three questions you have on a piece of paper. Then turn this into me at the beginning of class. (This will help you read the book more actively, and help me know what's confusing.) See also below. (You may skip up to four of these without penalty.)

### Students with disabilities

Students with disabilities requiring special accommodation should contact Office of Disability Services (ODS) promptly to discuss appropriate arrangements.

### Missed exams

If you have a conflict with any of the exam dates, you must contact me ahead of time so we can make arrangements. (At least a week ahead is preferable.) If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can.

## Syllabus and schedule.

Note: "+" indicates material not discussed in the textbook. Material in parentheses will probably be omitted from class (but discussed in problem sets).

Date Material Textbook Announcements
09/02 Sets, maps, vector spaces. §1.1-1.2, 2.1, 2.2 Welcome to V2020.
09/04 Subspaces, linear independence, span, bases and dimension. §2.3, 3.1, 3.2
09/09 Linear transformations, kernel, image. Matrices for linear transformations. §4.1, 4.2 Problem set 1 due.
09/11 Matrices for rotations, projections, (reflections) in dimensions 2 and 3. §4.5, +
09/16 Composition of transformations, matrix multiplication. §5.1, 5.6 Problem set 2 due.
09/18 Change of basis, similarity of matrices. +
09/23 Rank, row reduction, (elementary matrices), inverting a matrix. §5.2, 5.3, 5.5 Problem set 3 due.
09/25 Systems of linear equations. §7.1, 7.3, 7.5
09/30 Review   Problem set 4 due.
10/02 First midterm
10/07 Determinants and minors. §6.1,6.2,6.3
10/09 More on determinants. §6.4,6.5
10/14 Eigenvalues and eigenvectors. §9.1 Problem set 5 due.
10/16 The characteristic polynomial and diagonalization. §9.1,9.2+

10/21

Markov processes and the pagerank algorithm.

David Austin column on pagerank.

Brian White's lecture notes on pagerank (PDF).

Problem set 6 due.
10/23 Jordan normal form. §11.3+
10/28 Jordan normal form continued. +
10/30 The matrix exponential and systems of linear differential equations with constant coefficients. + Problem set 7 due.
11/4 Election day -- go vote.
11/6 Second midterm
11/11 Finishing up JNF. +
11/13

Inner products, inner product spaces.

§8.1, 8.2+
11/18

Hermitian inner products. Applications of inner products (correlation, least squares approximation). (Fourier series.)

Orthogonal transformations, unitary transformations, groups of matrices.

§8.3, 8.4

Problem set 8 due.
11/20

Symmetric matrices, Hermitian matrices, self-adjoint transformations. Statement of the principle axis theorem. Some reasons to care.

§10.1, 10.2, 10.3
11/25 Proof of the principle axis theorem. + Problem set 9 due.
11/27 Thanksgiving holiday.
12/2

Statement of the spectral theorem for orthogonal matrices. Statement of the general spectral theorem. (Proof of the spectral theorem.)

Quadratic forms, Sylvester's law.

§11.5+
12/4 Review   Problem set 10 due.

## Other advice.

A note on the textbook. Jänich's book, though pleasantly written, has relatively few examples. We will do more examples in class, but you might also find it helpful to have another book at hand as a cross reference. Almost any standard linear algebra text will do -- these can be had easily from the library, and used, old editions tend to be cheap to buy.

Reading mathematics. You are expected to read the sections in the textbook before coming to class. It's usually only a few pages, so read it carefully. Note down the questions you have; I would expect you to have at least one per page. Choose three of them and turn them in as the "active reading check." Read the section again after class. See which questions you now understand. Think about the remaining questions off and on for a day. See which you now understand. Ask someone (e.g., me) about the questions you still have left.

Linguistics and mathematics. I will make a big deal about using words in the course in a grammatically correct way. For example, for a vector space V, one says "a basis for V" but "the dimension of V." This will help you: often the grammar prevents you from making statements which are incorrect (or meaningless). So pay attention to it.

Getting help. If you're having trouble, get help immediately. Everyone who works seriously on mathematics struggles. But if you don't get help promptly you will soon be completely lost. The first places to look for help are my office hours and the course TA in the help room. Talking to your other classmates can also be helpful.

Teaching to learn. The best way to learn mathematics is to explain it to someone. You'll find that, particularly in office hours, I'll try to get you to explain the ideas. You should also try explaining the material to each other. The person doing the explaining will generally learn more than the explainee. Another thing to try is writing explanations to yourself, in plain English or as close as you can manage, of what's going on in the course. File them somewhere, and then look back at them a few days later, to see if your understanding has changed.