Outline of the course
Math 221-101
September 7, 2010
1
Office hours and contact information
Jonathan Wise
Office: MATH 209
e-mail: jonathan@math.ubc.ca
Office hours: Mondays and Wednesdays, 9—10:30; location TBA
Course webpage: http://www.math.ubc.ca/~jonathan/math221
2
Textbook
The official textbook is the UBC edition of Linear algebra and its applications
by David C. Lay. The UBC edition appears to be a pared down version of the
3rd edition, with some sections omitted.
I will try to make the course compatible with earlier editions of the textbook.
The publisher tries to make this difficult by renumbering the sections, so if you
are working with an earlier edition of the textbook, it’s a good idea to make
friends with someone who has the official version in order to check that you are
studying the right sections and doing the right exercises.
3
Introduction
The techniques of linear algebra are useful for solving a variety of problems that
may at first seem very different. Here is a sample:
1. (solving linear equations) Find all numbers x and y such that
3x + 4y = 7
(∗)
2x + y = 3.
2. (linear dynamics) Imagine two people, A and B, such that at each turn A
keeps 31 of his money and gives 13 to B, and B keeps 21 of his money and
gives
1
4
to A (the rest goes to waste). In equations:
1
1
An + B n
3
4
1
1
= An + B n .
3
2
An+1 =
Bn+1
Suppose that A starts with 100 dollars and and B starts with 50 dollars.
How much will A and B each have after playing for 1000 turns? (This kind
of problem is of great importance in many settings, including economics,
ecology, and even Google PageRank.)
3. (least squares approximation) Find the numbers (a, b, c, d) that make the
equation
ax + by + cz + d = 0
describe the plane that comes closest to passing through the four points
(1, 1, 1), (1, 0, 0), (1, 0, 3), and (2, −1, 1). (This kind of problem is closely
related to regression: approximating messy datasets by simple functions.)
What these examples have in common is linearity: variables are scaled by
constants and added together, but two variables are never multiplied together,
and variables are never exponentiated or used in exponents. You may already
know from experience that linearity is a very rare phenomenon: things are
usually much more complex. However, more complex phenomena can usually
be approximated linearly, which will allow you to use all of the tools we learn
in this course in a wide variety of settings.
How do you find a linear approximation to something more complicated?
That’s calculus!
4
Syllabus
Here are the main topics of this course:
1. the language of matrices and linear algebra (Chapters 1 and 2)
2. determinants, eigenvalues and eigenvectors (Chapter 5)
3. inner products (Chapter 6).
See the Google calendar on the course website for a (tentative) lecture-bylecture schedule.
5
Prerequisites
The only prerequisite is comfortability with high school algebra. If you can solve
a system of equations like Equations (∗) above, then you are ready to go. If you
have trouble solving Equations (∗) then you should do some review.
2
For the first half of the course, we will only need arithmetic and a small
amount of algebra. In the rest of the couse, we will use more algebraic tools.
You should be able to factor quadratic polynomials and have some idea about
how to manipulate higher degree polynomials.
There is also one point in the course where some familiarity with complex
numbers might come in handy. It’s not a prerequisite, though, and we’ll review
what we need when it comes up.
6
Succeeding in this course
Learning math is an active process: you can’t do it by just attending lecture
and reading the textbook. When new concepts are introduced in class or in the
text, you have to integrate those concepts with what you already know.
The best way to do that is to work exercises. You will submit exercises
making up 10% of your grade, but don’t assume that those are the only exercises
you should do. The more exercises you do, the better you will understand the
material, so try some of the additional problems that I suggest — or just pick
some problems on your own.
Your classmates are your greatest asset in this course, so get to know them
and work together on your homework assignments. I’m a crusty old teacher
and I’ve forgotten what it’s like to learn these things for the first time. Your
classmates are learning with you, and they will have fresh explanations that will
make sense when mine don’t. And explaining things to others can be a great
way to improve your own understanding.
There are also some specific topics that you should “grok”:
• row reduction (Section 1.2)
• matrix operations (Section 2.1).
The rest of the course will build on these, so please (!) put a special effort into
understanding them thoroughly.
7
Evaluation
Here is how your final mark will be calculated:
Homework
Midterm 1
Midterm 2
Final
10
20
20
50
%
%
%
%
Oct. 8
Nov. 19
TBA
I will drop your lowest homework score.
When writing up solutions, either on an exam or a homework assignment,
be as neat as possible and explain your reasoning clearly. Being neat means,
among other things, writing legibly, showing your work as a clear sequence of
3
steps from beginning to end, and stapling work that spans multiple pages. It
is a good idea to include some English words to explain what you are doing at
each step. A good rule of thumb is to write your solution as if you are explaining
it to one of your classmates.
You should plan to write every assignment twice: after solving all of the
problems, you should write up your work again to turn in.
Marks will be deducted if your work is not neat!
Remember: it is your responsibility to explain your solution clearly to the
grader, not the grader’s responsibility to divine your thought process! If the
grader can’t understand what you are doing, then what you are doing is wrong
and will be marked that way.
8
Attendance
Attendance is up to you, but you are responsible for everything that happens
in class. If you miss a class, make sure to get notes from a classmate.
If you miss an exam, you get a zero unless you’ve made arrangements with
me beforehand or have a note explaining that you could not take the exam
because of unforeseen circumstances. Your note really should explain that you
could not take the exam: if you were sick, your doctor’s note should say that
you were too sick to take the exam, not just that you were sick. Bring me the
note at the first opportunity.
Provided that you have a reasonable excuse, we’ll make arrangements for
the missed exam based on the circumstances.
4