Math 307 — Introduction to Linear Algebra — Fall 2013

Here is a shortcut to the course schedule/homework page.

**Lectures:** MTWF 11-11:50am in PM 124
**Office Hours:** MW2-2:50pm, T12-12:50pm, Θ11am-12:50pm, or by
appointment

**Instructor:** Jonathan
Poritz
**Office:** PM 248
**E-mail:**
jonathan.poritz@gmail.com

**Phone:** 549-2044 (office — any time); 357-MATH
(personal;please use sparingly)

**Text:** * Linear Algebra, A Modern Introduction
(3^{rd} edition)*, by David Poole (the same
textbook as was used in Math 207).

**Prerequisites:** A satisfactory grade (C or higher) in Math 207
(Matrix and Vector Algebra) and Math 224 (Calculus II): 207 because 307
builds directly on the material of 207; 224 is required as part of a general
attempt to enforce a level of mathematical maturity of students enrolled in
307, and to synchronize correctly the multivariate calculus with courses
teaching vectors and matrices. The course catalog also says "knowledge of
a programming language" is required, but I will be somewhat flexible in this
regard — please contact me individually if you have concerns about this.

**Postrequisites:** This course is a prerequisite for

- Math 421 Introduction to Analysis,
- Math 427 Abstract Algebra, and
- Math 445 Discrete Mathematics;

- Math 319 Number Theory,
- Math 463 History of Mathematics, and
- Math 477 Materials and Techniques of Teaching Secondary School Mathematics.

**Course Content/Objective:** The Catalog says simply that this course is

There are actually two important pieces here:A rigorous development of vector spaces and linear transformations.

** A rigorous development** means that this course is designed to be
an introduction to reading and writing clear, correct, logical, unambiguous,
and formal definitions, theorems and proofs. This is in many ways the final
frontier of your mathematics education — to get to this point, you have
learned how to work with numbers, variables, operations, equations, functions,
figures/graphs, and algorithms; now, we will work on making precise

The rigor and abstraction of the statements and proofs which will be the center
of this course can be a challenge to students. A pessimist might say we are
now turning all of mathematics into word problems, and moving away from the
thing students are best at, by this point in their mathematical careers:
calculating stuff. A pragmatist would reply, however, that the abstraction is
an approach that vastly broadens the applicability of our work (in other
areas of mathematics and in all applied fields), and that two and a half
thousand years of evidence shows the power of this approach. An optimist would
go on to add that there is much more *beauty* in a well-crafted theorem
statement or proof than in a mere calculation, and an enormous opportunity for
*creativity* that following some plug-and-chug algorithm from an earlier
class would never allow.

** Vector spaces and linear transformations** are the other piece
mentioned in the above catalog description. The examination of these
mathematical structures will be the particular domain in which we will
develop our skills of abstraction and proof. A vector space is one of the
most fundamental mathematical objects, being built out of only two operations
(vector addition and scalar multiplication) that satisfy a few simple
properties. And when working with vector spaces, the most natural functions
to use are those which preserve these two operations, which functions are
then called linear transformations.

*Linear algebra* per se is then the basic manipulation and understanding
of these spaces and transformations, and its abstract and general results
yield powerful, concrete consequences wherever they are applied. It is used
across all of pure and applied mathematics, but also in physics, chemistry,
mathematical economics and sociology, computer science, engineering... the
list goes on and on. Our second goal in this course, then, is to master a
good piece of this theory, in its shimmering, abstract perfection, and also
to see how it can be applied in just a few of the myriad possible ways.

**Academic integrity:** Mathematics is more effectively and easily
learned — and more fun — when you work in groups.
However, all work you turn in must be your own, and any form of cheating
is grounds for an immediate **F** in the course for all involved parties. In
particular, some assignments, such as take-home portions of tests, will have
very specific instructions about the kinds of help you may seek or resources
you may use, and violations of of these instructions will not be tolerated.

**Attendance, work ratio, and classroom participation [Miniquizzes]:**
Regular attendance in class is a key to success. But more than merely
attending, you are also expected to be *engaged* with the material in
the class. In order for this to be possible, it is necessary to be current
with required outside activities such as reading textbook sections, thinking
about problems, doing the small writing assignments and larger problem sets.
You are expected to spend 2-3 hours per hour of class on this outside work
— this is not an exaggeration (or a joke!), in fact it is closer to a
legal requirement. To encourage you to stay in synch with this outside work,
so you will be able to get the most out of class time, **the majority of
class meetings will include a miniquiz**. Your lowest five miniquiz
scores will be dropped.

If you absolutely have to miss a class, please inform me in advance (as late as a few minutes before class by phone or e-mail would be fine) and I will video the class and post the video on the 'net. You can then watch the class you missed in the comfort of you home and (hopefully) not fall behind. Classes I have videoed will have the icon next to that day's entry on the schedule/homework page to remind you of the available video.

**Homework:** Mathematics is not a spectator sport, it is something you
** do**. You would not expect to learn a musical instrument, or
prepare for an athletic event, by watching someone else play that instrument
or do that event. The statements and examples we discuss in class or you see
in the book will lie there like inanimate, two-dimensional ink on the page or
chalk on the board until you breathe full, three- (or more!) dimensional life
into them with your insight and imagination ... by working through them at
your own pace, on your own, and applying them in problem-solving. There will
be plenty of opportunity to exercise these creative talents in class, but you
will need to work extensively outside of class to practice and refine them.
This will take the form of exercises sets you will work on and hand in every
few days. We will not have large, weekly problem sets in this class: instead,
we will have small sets due roughly every other class. This way the sets will
not be individually too onerous, and keeping up with them will be another way
to stay in synch with the classroom activity.

Of course, a significant part of the homework in this class will consist of writing proofs, which is a task that requires actual creative insight, unlike a straight computation. Now one of the things about creativity is that it does not like to be rushed, so it is a very bad idea to expect to be able to slam out a proof in a short time. I therefore highly recommend that you start working on a HW set as early as possible, so your muse of creativity has time to visit you, no matter what her schedule is that week.

Some organizational details about homework:

- Homework is due either in class or at my office,
**no later than 3pm**. - Homework is assigned by day but graded by problem. Each problem will
typically be worth
**3 points**, meaning:- problem entirely missing (or wrong problem done!)
- some work present, but also several errors and/or important missing parts;
- most of the correct content is present, but there is at least one key idea or step which is missing, and/or there is a significant flaw in exposition (a variable used without definition, that kind of thing);
- all content is present, all notation is defined, all steps are explained and justified.

- Your ten lowest HW scores (on individual problems) will be dropped.
- Homework problems will appear on the homework web page on a regular basis. Please get used to going to that page frequently — at least every other day, and certainly before starting your work on a homework set.
- Late homework will count, but at a reduced value — generally, the score will be reduced by around a point for each day late.
- Please try to be neat (how can I give you credit for your work if I
cannot read it?). In particular, don't skimp on paper! If possible,
cut off ragged edges and use staples to attach multiple pages (not that
terrible thing where you sort of chew on the corner of the pages). But
I care much more about the
**content**than the**form**of your work, so don't worry if you just have bad handwriting or something (I certainly do!). - Make sure to label each assignment you hand in with your name and date, the course number, and number of the homework assignment (from the HW page); if you are handing in something which is not stapled, please write your name on each page.
- I am trying to reduce the carbon footprint of my classes. So I ask that
you reuse paper whenever possible, by taking any pages you can find that
are blank on one side (handouts from other classes, drafts of your work
for this or other classes,
*etc.*), putting a big "X" over the previously used side, and doing your HW for this class on the blank side. To encourage this, I will keep track of how many such reused pages you hand in and they will be worth some*green points*extra credit at the end of the term (probably one point every ten pages or so).

**Maxiquizzes:** Most Fridays, during weeks in which there is no hour
exam, there will be a short (≈15 minute) quiz (which we will call a
*maxiquiz*, to contrast with the daily miniquizzes) at the end of class.
These will be graded out of 10; your lowest two quiz scores will be dropped.

**Journal:** Students must keep a careful, complete list of all of the
definitions and theorems (also lemmata and corollaries, *etc.*) which we
discuss in class, in the form of a class *journal*. The goal of this
journal is simply to be a to keep in one place all the formal terminology and
all results, techniques, and facts which you would want to use to study for a
quiz or test, or to consult when doing homework. For example, every single
miniquiz should be completely trivial if you were to do it with your journal
in front of you.

Journals will be graded out of **5 points**: you should expect to get mostly
perfect scores (you can create them on your own time, in consultation with your
class notes and the textbook!). If there is something important missing from
the journal, or if it has something which is simply incorrect (there will be a
higher standard for exact correctness in journals than in, say, the homeworks,
since you may be studying from your journal or referring to it while doing
homework), you may loose some points. But you can earn all of those points back
by handing in a corrected journ within the next three class days.

**Exams:** We will have two midterm exams on dates to be determined (and
announced at least a week in advance). These may have a take-home portion in
addition to the in-class part. Our **final exam** is scheduled for both
**Tuesday, December 10th** and **Wednesday, December 11th**,
**from 10:30am to 12:50pm in our usual classroom** on both days.

**Revision of work on homework, quizzes, and tests:** A great learning
opportunity is often missed by students who get back a piece of work graded by
their instructor and simply shrug their shoulders and move on. In fact,
painful though it may be, looking over the mistakes on those returned papers
is often the best way to figure out exactly where *you* tend to make
mistakes. If you correct that work, taking the time to make sure you really
understand completely what was missing or incorrect, you will often truly master
the technique in question, and never again make any similar mistake.

In order to encourage students to go through this learning experience, I will
allow students to hand in revised solutions to all homeworks, maxiquizzes, and
midterms. There will be an expectation of slightly higher quality of
exposition (more clear and complete explanations, all details shown, all
theorems or results that you use carefully cited, *etc.*), but you will
be able to earn a percentage of the points you originally lost, so long as you
hand in the revised work at the very next class meeting. The percentage you
can earn back is given in the "revision %" column of the
table in the **Grades** section, below.

**Grades:** On quiz or exam days, attendance is required — if you
miss a quiz or exam, you will get a **zero** as score; you will be able to
replace that zero only if you are regularly attending class and have informed
me, **in advance**, of your valid reason for missing that day.

In each grading category, the total points possible will be multiplied by a
normalizing factor so as to come to 100. Then the different categories will
be combined, each weighted by the "course %" from the following table, to
compute your total course points out of 100. Your letter grade will then be
computed in a manner not more strict than the traditional "90-100% is an
**A**, 80-90% a **B**, *etc.*" method. *It is the policy of the
Department of Mathematics and Physics that fractioned course grades (plusses
and minuses) will NOT be assigned.*

pts each | # of such | # dropped | revision % | course % | |
---|---|---|---|---|---|

Miniquizzes: | 2 | ≈45 | 5 | 0% | 8% |

Maxiquizzes: | 5 | ≈12 | 2 | 50% | 9% |

Homework: | 3/prob | ≈20 sets ≈100 probs |
10 probs | 75% | 25% |

Journal: | 5 | ≈15 | 1 | 100% | 8% |

Midterms: | >100 | 2 | 0 | 33.3% | 25% |

Final Exam: | >200 | 1 | 0 | 0% | 25% |

** Nota bene:** Most rules on due dates, admissibility of make-up
work, etc., will be interpreted with great flexibility for students who are
otherwise in good standing (

**Contact outside class:** Over the years I have been teaching, I have
noticed that the students who come to see me outside class are very often the
ones who do well in my classes. Now correlation is not causation, but why not
put yourself in the right statistical group and drop in sometime? I am always
in my office, PM 248, during official office hours. If you want to talk to me
privately and/or cannot make those times, please mention it to me in class or
by e-mail, and we can find another time. Please feel free to contact me for
help also by e-mail at
jonathan.poritz@gmail.com, to
which I will try to respond quite quickly (usually within the day, often
much more quickly); be aware, however, that it is hard to do complex
mathematics by e-mail, so if the issue you raise in an e-mail is too hard
for me to answer in that form, it may well be better if we meet before the
next class, or even talk on the telephone (in which case, include in your
e-mail a number where I can reach you).

**A request about e-mail:** E-mail is a great way to keep in touch
with me, but since I tell all my students that, I get *a lot* of e-mail.
So to help me stay organized, please **put your full name and the course name
or number in the subject line of all messages to me**. Also, if you are
writing me for help on a particular problem, please do not assume I have my
book, it is often not available to me when I am answering e-mail; therefore,
you should give me enough information about the problem so that I can
actually help you solve it (*i.e.,* "How do you do problem number
*n* on page *p*" is often not a question I will be able to answer).

**Students with disabilities:** This University abides by the Americans
with Disabilities Act and Section 504 of the Rehabilitation Act of 1973, which
stipulate that no student shall be denied the benefits of education "solely by
reason of a handicap." If you have a documented disability that may impact your
work in this class for which you may require accommodations, please see the
Disability Resource Coordinator as soon as possible to arrange accommodations.
In order to receive accommodations, you must be registered with and provide
documentation of your disability to: the Disability Resource Office, which is
located in the Library and Academic Resources Center, Suite 169.

A is an m×n real matrix |
A^{T}, the transpose of A, is n×m |

K is the nullspace of A |
L is the nullspace of A^{T} |

R is the rowspace of A |
C is the columnspace of A |

It is true that, as indicated in the picture,

If

(which is nothing other than

since

If

then in fact: