This course ended
on December 9, 2020 |

## Math 204: Linear Algebra

Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall 2020. Instructor: David J. Eck (eck@hws.edu) Syllabus: http://math.hws.edu/eck/courses/math204_f20.html Textbook PDF: Linear Algebra, 4th edition, by Jim Hefferon Supporting materials at https://hefferon.net/linearalgebra Monday, Wednesday, Friday, 9:50–10:50 AM Gearan Center 102: Froelich Recital Hall

Homework | |||

Homework 1, Due August 31, |
Homework 2, Due September 9, |
Homework 3, Due September 16, |
Homework 4, Due September 24, |

Homework 5, Due October 1, |
Homework 6, Due October 13, |
Homework 7, Due October 20, |
Homework 8, Due October 29, |

Homework 9, Due November 10, |
Homework 10, Due December 1, |
Final Exam, Due 8:30 AM, December 9, |

### Final Exam: December 9

The official final exam period for this course is 8:30 to 11:30 AM on Wednesday, December 9. Our exam is a take-home test that is due by the start of that final exam period. The exam, with full instructions, is available here.

### Fifteenth Week: November 30 and December 2

The final two classes of the semester will be online, using the usual Zoom meeting link for Math 204, which can be found on the Canvas page for this course. These classes will look at some applications of matrices and eigenvalues, including solutions of first-order linear systems of differential equations and Markov processes.

The final homework assignment is due on Tuesday, December 1.

### Fourteenth Week: November 23

We have our final in-person class of the semester on Monday, before breaking for Thanksgiving. After the break, we will be back for a couple online classes to end the semester.

We have finished all of the material that we really needed to cover in this course. In the remaining time, I will add some quick introductions to several additional topics. On November 23, I will talk about products, quotients, and direct sums of vector spaces.

Here is my very short summary of the material from last week, posted a little late:

### Thirteenth Week: November 16, 18, and 20

This week, we will be looking at selected material from Chapter 5 on eigenvalues, eigenvectors, diagonalizability, and similarity of matrices. This material is in Chapter Five, Section II. I might state the Jordan Canonical Form theorem, but we won't do anything else from Chapter Five, Sections III and IV.

### Twelfth Week: November 9, 11, and 13

There is an in-class test this Friday, November 13. A study guide is available.

Any new material that we cover this week will not be on the test. On Monday, we will start looking at complex numbers and complex vector spaces. Here is a short summary of what you will need to know about this:

### Eleventh Week: November 2, 4, and 6

We will work on Chapter Four this week. The topic is determinants. We will see how to define and how to calculate the determinant of an n×n matrix, and we will study various properties of the determinant. We will cover every part of Chapter Five. Most of what will do is from Sections I.2, II.3, and III.1. The hope is to finish entirely with determinants this week. Some of the major results are covered in this guide:

Remember that there is a test coming up next week on Friday, November 13.

### Tenth Week: October 26, 28, and 30

**Added on October 30:** Here are links to the two programs that I used in class today
to demonstrate how affine maps can be used to make fractals:

Sierpinski Demo (Not a polished program!!)

Choosing bases for finite-dimensional vector spaces allows us to represent vectors as column vectors in
**R**^{n}
and homomorphisms as n×m matrices.
We start this week by looking at what happens to these representations of vectors and
homomorphisms when the choice of basis is changed. This material is from Chapter Three, Section V
in the textbook. We will not cover the option section, Chapter Three, Section VI.
Before we go on to Chapter Five, I am planning to introduce some material on "affine transformations,"
which I have already mentioned in class. An affine transformation on **R**^{n}
can be written as a linear transformation followed by a translation. Here are my short guide to the
basics of change of basis and of affine transformations:

### Ninth Week: October 19, 21, and 23

We have looked at how multiplication of vectors in **R**^{n}
by an n×m matrix defines a homomorphism from **R**^{n} to
**R**^{m}. And we have seen that matrix multiplication corresponds
to composition of homomorphisms. That material is covered in the textbook
in Chapter Three, Section IV. This week, we return to Chapter 3, Section III
to look at matrix representations for general homomorphisms between finite-dimensional
vector spaces. My guide for this material is here (but I have to say that
my "guides" have become more sketchy as the semester has progressed):

We will also finish some left-over material from Chapter 3, Section IV. In particular, we will see how row operations can be represented by matrices.

### Eighth Week: October 12, 14, and 16

We will continue our study of homomorphisms. First, we will finish up Chapter Three, Section II.2
by looking at the range space, null space, rank, and nullity of a homormorphism between finite-dimensional
vector spaces. After that, I will be departing somewhat from the order of material in the textbook.
The book covers the representation of linear maps by matrices in a very general way. I think that
it will be less confusing to cover maps from **R**^{n} to **R**^{m}.
This will require doing some of the material from Chapter Three, Section IV before returning
to Chapter Three, Section III. Here is a short guide to matrix operations and matrix representation
of homomorphisms from **R**^{n} to **R**^{m}:

### Seventh Week: October 5, 7, and 9

We have one more section to do in Chapter Two: Section III.3, which introduces the "rank" of a matrix. We will not cover the optional section III.4 in Chapter Two.

We will then move on to Chapter Three, which covers the fundamental concept of
"homomorphism" or "linear map" between vector spaces. A homomorphism from a vector
space V to a vector space W is simply a function from V to W that preserves the
vector space operations: f(u+v) = f(u)+f(v) for u,v in V, and f(rv) = rf(v) for
r in **R** and v in V. A homomorphism that is a bijection is an "isomorphism."
The textbook introduces isomorphisms in Chapter Three, Section I and it introduces
general homomorphisms in Chapter Three, Section II. We should cover Section I and
begin Section II this week. However, we will do some facts about homomorphisms in
general, before turning to isomorphisms.
Short guides to this week's material can be found here:

You can expect the test from last Friday to be returned on Wednesday.

### Sixth Week: September 28 and 30; October 2

A test is planned for Friday, October 2. An information sheet is available. The test will be given in person in class. If it is not possible for you to be in class, you should let me know.

Aside from the test, we will be finishing up Chapter Three, Section III.2, which is part of the material for the test. That section covers dimension of finite-dimensional vector spaces and the fact that every basis of a finite-dimensional vector space has the same number of elements. We will do some review for the test, and there will certainly be time for questions. If we have extra time, we will move on to new material from Chapter Two, Section III.3.

Remember that Homework 5 is due by **noon** on Thursday, October 1.

### Fifth Week: September 21, 23, and 25

We ended last week by introducing linear independence. We will continue
that this week, and we will look at the idea of a **basis** of a vector
space and the **dimension** of a vector space. A basis for a vector
space is simply a subset of that vector space that spans the entire vector
space and is linearly independent. This is one of the central concepts
in linear algebra. The reading is Chapter Two, sections II.1, III.1,
and III.2.
Here is a reading guide:

### Fourth Week: September 14, 16, and 18

After a brief remark on accuracy of computation, which was promised for last week, we move on to Chapter 2, which introduces vector spaces. We have been working with vectors as columns or rows of numbers, but the general concept of vector space is more general and more abstract. We will look at the definition and some basic properties of vectors spaces, subspaces, spans, and linear combinations. The reading guides for this week are

### Third Week: September 7, 9, and 11

We will finish up Chapter One this week. The reading is Chapter One, Section III. The main topic is reduced row echelon form, but we will also be looking to get a better handle on the meaning of linear combination and how to work with it. Along the way, we will need to go over the idea of equivalence relation and the proof technique of mathematical induction. We will also be looking briefly at one of the topics at the end of the chapter, "Accuracy of Computation." For the most part, we will not cover any of the end-of-chapter topics, but I believe that it is important to be aware of the complications that arise from the fact that numerical computation and measurement are not exact.

Homework 2 is due on Wednesday. You should plan to turn in homework on time. If circumstances make that difficult for you, you should consult with me about getting an extension. Note that sample Homework 1 solutions are available. The reading guides for this week's material are:

### Second Week: August 31; September 2 and 4

For the first part of the week, we will be talking about the geometry of vectors and linear systems. I am putting more emphasis on the geometry than the book does, so some of this material is not covered in the book, or is covered later. We will also be looking at "homogeneous" linear systems and how they relate to linear systems in general. The reading from the book is Chapter One, Section I.3. There are two "reading guides" for this material:

Here is example shown in class on Monday: August 31 example

My office hours on Zoom this week will be Tuesday and Thursday, 12:30 to 2:00 and 6:30 to 8:00. You can use the Canvas Calendar feature to make an appointment during those times. Note that I will usually be in my office between classes on Monday, Wednesday, and Friday, from about 11:15 to 1:00. If you want to meet either on Zoom or in person during those times, you should arrange an appointment by email or by talking to me after class, or you should call my office (315-781-3398) to see whether I am available.

### First Week: August 24, 26, and 28

Added Aug. 27: A LaTeX example from Wednesday's class and the compiled PDF.

Welcome to the course!

You should download the PDF of the textbook and start reading Chapter One. We will try to cover most of
Sections I.1, I.2, II.1, and II.2 from that chapter this week, although we will continue that
material into the first part of next week. The main topics are Gauss's method for
solving systems of linear equations and vectors in **R**^{n}. You are probably already familiar with the general
idea of solving linear equations, but this is a good starting point for a course in linear algebra. And the
sections on vectors are central to the course. I will also spend some time in class talking about
the LaTeX typesetting system.

You should also carefully read the syllabus for the course!

The PDF version of the textbook is free. You will not need a printed copy, but if you would like to have one, you can order a copy from amazon.com through this link.

The first homework assignment is already available. You are encouraged to write up your solutions in LaTeX, using a free account at overleaf.com, but I will also accept handwritten work scanned to a PDF file, at least at first. For more information about submitting homework, see

http://math.hws.edu/eck/math204/f20/submitting-homework.html

I plan to post short "reading guides" for most lectures, which can be found at this link. The first three installments are

The third installment will take us into the beginning of the second week.