Colorado State University — Pueblo, Fall 2015
Math 307, Introduction to Linear Algebra

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Lectures: MTWF 10:10-11:05am in PM 116      Office Hours: MTWF 11:15am-12:10pm and T 9:05-10:00am, or by appointment

Instructor: Jonathan Poritz     Office: PM 248     E-mail:
Phone: 549-2044 (office — any time); 357-MATH (personal; please use sparingly)

Text: Linear Algebra, A Modern Introduction (3rd 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.

Postrequisites: This course is a requirement for the mathematics major. It is a prerequisite for

and is a possible prerequisite, along with the alternative Math 320 Introductory Discrete Mathematics, for

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

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

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 statements about mathematical situations, for which we will need precise definitions of mathematical concepts, and on making irrefutable proofs of these statements. Along the way, we will need to discuss what makes a good definition or statement of a result and what is a complete and careful proof. Some rudiments of mathematical logic will be developed, along with convenient meta-mathematical terminology and notation.

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.

Attendance, work ratio, and 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, nearly every class meeting [when there is no maxiquiz or test] will include a miniquiz, usually simply to state a current definition or theorem. 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 Black and white camera icon next to that day's entry on the schedule/homework page to remind you of the available video.

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. Problems will be similar to those on the homework. These will be graded out of 10; your lowest two quiz scores will be dropped.

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 exercise 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:

Important Idea Indices [I3s]: With each homework set you hand in, you should include, on a separate paper, a succinct but complete statement of the most important ideas since the last I3. Typically, this will be a list of the definitions and theorems (also lemmata and corollaries, etc.) which we discuss in class, were in the reading, or which you used when doing the homework problems. The goal of these I3s is to put together in one place all the formal terminology, results, techniques, and facts which you would want to use to study for a quiz or test and to consult when doing homework. For example, every single miniquiz should be completely trivial if you were to do it with your collected I3s in front of you. In fact, putting them all together at the end of the semester should result in a complete study guide for the final.

I3s 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 a particular I3 you hand in, or if it has something which is simply incorrect (there will be a higher standard for exact correctness in I3s than in, say, the homeworks, since you may be studying from your I3s or referring to them while doing homework), you may loose some points. But you can earn all of those points back by handing in a corrected I3 by the next class meeting.

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 Thursday, December 10th from 8:00am-10:20am and Friday, December 11th from 10:30am-12:50m, 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 [e.g., by e-mail], 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% 10%
Maxiquizzes: 10 ≈12 2 50% 10%
Homework: 3/prob ≈20 sets
≈100 probs
10 probs 75% 20%
I3s: 5 ≈20 1 100% 10%
Midterms: >100 2 0 33.3% 25%
Final Exam: >200 1 0 0% 25%
Green Points: 1/page ≤200 ? 0 0% XC

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 (i.e., regular classroom attendance, homework (nearly) all turned in on time, no missing quizzes and tests, etc.) when they experience temporary emergency situations. Please speak to me — the earlier the better — in person should this be necessary for you.

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, 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).

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.

Accommodations: 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.


Here's a diagrammatic representation of something sometimes called

The Fundamental Theorem of Linear Algebra:

A is an m×n real matrix    AT, the transpose of A, is n×m
K is the nullspace of A    L is the nullspace of AT
R is the rowspace of A    C is the columnspace of A

It is true that, as indicated in the picture,   KR  and  LC .

If r = rank(A), then dim(R) = r = dim(C)n = dim(K) + r, and m = dim(L) + r
  (which is nothing other than The Rank-Nullity Theorem,
since dim(K) = nullity(A) and dim(L) = nullity(AT)).

If TA represents multiplication on the left by A and likewise for TAT,
then in fact: TA : R ≅ C,  TAT : C ≅ R,  K =ker(TA), and L =ker(TAT).


Jonathan Poritz (
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