## Math 204: Linear Algebra

Department of Mathematics and Computer Science Hobart and William Smith Colleges Spring 2020. Instructor: David J. Eck (eck@hws.edu) Syllabus: http://math.hws.edu/eck/courses/math204_s20.html Textbook PDF: Linear Algebra, 3rd edition, by Jim Hefferon Supporting materials at http://joshua.smcvt.edu/linearalgebra/

Homework | |||

Homework 1 Due Jan. 29 Answers |
Homework 2 Due Feb. 5 Answers |
Homework 3 Due Feb. 14 Answers |
Homework 4 Due Feb. 19 Answers |

Homework 5 Due March 2 Answers |
Homework 6 Due March 13 |

### Ninth Week and Beyond...

Because of the COVID-19 pandemic, we are moving to entirely on-line teaching. The main hub for the course will be on Canvas. Students should check there for further updates and assignments.

### Eighth Week: March 9, 11, and 13

We have begun Chapter 3, which covers linear functions between vector spaces. In class, we have looked at the general idea of linear function and at how matrices can be used to define functions. However the text book begins the subject with isomorphisms and automorphisms. This week, we should finish Chapter 3, Section I and most if not all of Section II.

### Seventh Week: March 2, 4, and 6

We will start the week with the big proof left over from Friday: That if V is a finite-dimensional vector space, then every basis of V has the same number of elements. We will then cover Chapter 2, section III.3, which discusses the idea of rank of a matrix. We will skip Chapter 2, Section III.4, so we should start Chapter 3 by the end of the week. Chapter 3 is about functions between vector spaces and their relationship to matrices and matrix multiplication. I might begin with an overview of this material before continuing with the presentation in the book.

### Sixth Week: February 24, 26, and 28

The reading for the week is Chapter 2, Sections III.1 and III.2, covering the idea of basis of a vector space and the definition of dimension of a finite-dimensional vector space. We might start Section III.3 on Friday.

Part of the homework for this week will be to choose a presentation topic. A list of possible topics was handed out on the first day of class. Here is a link to that list:

### Fifth Week: February 17, 19, and 21

There is test this week on Friday, February 21. A study guide was handed out in class on Monday.

We started Chapter II, Section II last week, which covers linear independence. We will continue looking at linear independence and spanning sets on Monday, and that material will be coverd on the test. On Wednesday, I will answer any questions about the test. If there is extra time, we will begin Chapter 2, Section III, by looking at the idea of a basis for a vector space.

### Fourth Week: February 10, 12, and 14

At end of last week, we just started work on abstract vector spaces. and we will get into that topic more seriously this week. The reading is Chapter 2, Sections I.1, I.2, and II.1.

### Third Week: February 3, 5, and 7

We will cover sections III.1 and III.2 from Chapter 1 this week. The topic is reduced row echelon form. We will show that a given matrix has exactly one reduced row echelon form. (The proofs that we will do this week will require us to talk about equivalence relations and mathematical induction.) We will also take a look at some of the things that computer algebra systems can do for linear algebra.

### Second Week: January 27, 29, and 31

The reading for the week is Sections I.3, II.1, and II.2.
Section I.3 introduces *homogeneous* linear systems, which just means that all the
constant terms on the right are zero, and it proves something important about the structure
of the solution set of any system of linear equations. Sections II.1 and II.2 are about the
geometry of vectors in **R**^{n}, especially in **R**^{2} and
**R**^{3}. These sections cover things representing vectors as arrows,
the dot product of vectors, length of vectors, and orthogonality.
This material is marked as optional in the textbook, and it is not used all that
much in the rest of the course, but when working with vector spaces, it's a good
idea to understand how they work geometrically.

### First Week: January 22 and 24

Welcome to the course!

You should download the PDF of the textbook and start reading Chapter One. We will cover Sections I.1 and I.2 from that chapter this week. The main topic is Gauss's method for solving systems of linear equations. You are probably already familiar with the general idea, but this is a good starting point for a course in linear algebra.

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, the bookstore will have a few copies, or you can order a copy from amazon.com through this link. A copy will also be on reserve in the library.