MATHEMATICS 23a/E23a, FALL 2015 Linear Algebra and Real Analysis I
Syllabus for undergraduates and local Extension students
(Distance Extension students will also need to consult a special syllabus) Last revised: July 22, 2015
Course Website: https://canvas.harvard.edu/courses/4524 Instructor: Paul Bamberg (to be addressed as “Paul,” please)
Paul graduated from Harvard in 1963 with a degree in physics and received his doctorate in theoretical physics at Oxford in 1967. He taught in the Harvard physics department from 1967 to 1995 and joined the math department in 2001. From 1982 to 2000 he was one of the principals of the speech recognition company Dragon Systems. If you count Extension School and Summer School, he has prob-ably taught more courses, in mathematics, physics, and computer science, than anyone else in the history of Harvard. He was the first recipient of the White Prize for excellence in teaching introductory physics.
This term, Paul is also teaching Math 152, “Discrete Mathematics,” and Math 116, “Real Anaysis, Convexity, and Optimization.”
Email: [email protected]
Office: Science Center 322, (617-49)5-9560 Office Hours:
Tuesday and Thursday, 1:30-2:15 in Science Center 322. Mondays 2-2:30 (longer if students are still there)
Head Teaching Assistant: Kate Penner (to be addressed as “Kate,” please) Kate is the course head for Math E-23a, responsible for making it possible for students from around the nation and the world to participate as fully as possible in course activities.
Kate’s Harvard undergraduate degree is in government, but her interests have moved to political economy and mathematics. After taking Math E-23 in the Extension School, she became the head teaching assistant and is starting her sixth year in that position. She has been course head for linear algebra and real analysis courses in the Summer School. She may have set a Harvard record in Spring 2013 by teaching in four courses (Math M, Math 21b, Math 23b, and Math 117). To date, she has received over a dozen teaching awards from the Bok Center for Teaching and Learning for her work teaching undergraduate math.
This term, Kate is also teaching Math1a. Email: [email protected]
Office: Science Center 424 Office Hours: TBD
Course Assistants:(all former students in Math 23a or Math E-23a) • Nicolas Campos, [email protected]
• Jennifer Hu, [email protected] • Ju Hyun Lee, [email protected] • Elaine Reichert, [email protected] • Ben Sorscher, [email protected]
• Sebastian Wagner-Carena, [email protected] • Kenneth Wang, [email protected]
Goals: Math 23a is the first half of a moderately rigorous course in linear algebra and multivariable calculus, designed for students who are serious about mathemat-ics and interested in being able to prove the theorems that they use but who are as much concerned about the application of mathematics in fields like physics and economics as about “pure mathematics” for its own sake. Trying to cover both theory and practice makes for a challenging course with a lot of material, but it is appropriate for the audience!
Prerequisites: This course is designed for the student who received a grade of 5 on the Math BC Advanced Placement examination or an A or A minus in Math 1b. Probably the most important prerequisite is the attitude that mathematics is fun and exciting. Extension students should ordinarily have an A in Math E-16, and an additional math course would be a very good idea.
Our assumption is that the typical Math 23a student knows only high-school algebra and single-variable calculus, is currently better at formula-crunching than at doing proofs, and likes to see examples to accompany abstractions. If, before coming to Harvard, you took courses in both linear algebra and multivariable calculus, Math 25 might be more appropriate. We do not assume that Math 23 students have any prior experience in either of these areas beyond solving systems of linear equations in high school algebra.
This year, for the second time, we will devote four weeks to single-variable real analysis. Real analysis is the study of real-valued functions and their properties, such as continuity, and differentiability, as well as sequences, series, limits, and convergence. This means that if you are an international student whose curriculum included calculus but not infinite series OR if you had a calculus course that touched only lightly on topics like series, limits, and continuity, you will be OK.
Mathematics beyond AP calculus is NOT a prerequisite! Anyone who tries to tell you otherwise is misguided. In fact, since we will be teaching sequences and series from scratch (but rigorously), you can perhaps get away with a weaker background in this area than is required for Math 21.
Strange as it may seem, Part I of the math placement test that freshmen have taken is the most important. Students who do well in Math 23 have almost all scored 26 or more out of 30 on this part.
Extension students who register for graduate credit are required to learn and use the scripting language R. This option is also available to everyone else in the course. You need to be only an experienced software user, not a programmer.
Who takes Math 23?
When students in Math 23b were asked to list the two concentrations they were most seriously considering, the most popular choices were mathematics, applied math, physics, computer science, chemistry, mathematical economics, life sciences, and humanities.
Extension students who take this course are often establishing their credentials for a graduate program in a field like mathematical economics, mathematics, or engineering. Programs in fields like economics like to see a course in real analysis on your transcript. Successful Math E-23 students have usually taken more than one course beyond single-variable calculus.
Upperclassmen who have made a belated decision to go into a quantitative PhD program will also find this course useful.
Course Meetings:
The course ordinarily meets in Science Center A. To avoid overcrowding, the first two lectures have been moved to Science Center C.
Lectures on Tuesdays and Thursdays run from 2:37 to 4:00. They provide complete coverage of the week’s material, occasionally illustrated by examples done in the R scripting language.
Problem Sessions (Section)
There are two types of weekly problem sessions led by the course staff. The first is required; the second, though highly recommended, is optional.
• The “early” sections on Thursday and Friday will be devoted to problem solving in small groups. These are a required course activity and will count toward your grade. Lecture on Thursday is crucial background for section!
• The “late” sections that meet on Monday will focus on the weekly problem sets due on Wednesday mornings, and will also review the proofs that were done in lecture. Attendance at these sections is optional, but most students find them to be time well spent.
Videos will be made of all the lectures. Usually the video will be posted on the Web site before the next lecture, and often it will appear on the same day. The Thursday video will not be posted in time to provide preparation for the early sections that meet on Thursdays, and we cannot guarantee that it will appear before the Friday sections.
Even though all lectures are captured on video, Harvard rules forbid under-graduates to register for another course that meets at the same time as Math 23, even one with just a 30-minute overlap! Here is the official statement of this year’s policy:
“In recent years, the Ad Board has approved petitions in which the direct and personal compensatory instruction has been provided via video capture of classroom presentations. In keeping with the views of the Standing Committee
Council and the full faculty last April, the Ad Board will no longer approve such petitions.”
With regard to athletic practices that occur at the same time as classes, policy is less well defined. Here is the view of the assistant director of athletics:
”The basic answer is that our coaches should be accommodating to any aca-demic conflict that comes up with class scheduling. Kids should be able to take the classes they want and still be a part of the team. Especially for classes that would only cause a student to miss a small part of a practice.
What complicates things are the classes that would cause a student to miss an entire practice for 2-3 days a week. Those instances make it hard for a student to engage fully in the sport and prepare adequately for competition.
It’s hard for freshmen to ask a coach - the adult they have the closest relation-ship to in campus - for practice accommodations but in my experience many of them will work with students on their total experience”
The Math 23 policy, based on this opinion: It is OK to take Math 23a and practice for your sport every Tuesday, but you must not miss Thursday lecture for a practice.
Extension students may choose between attending lecture or watching videos. However, students in Math E-23a who will not regularly attend lecture on Thursday should sign up for a section that meets as late as possible. Then, with occasional exceptions, they can watch the video of the Thursday lecture to prepare for section. Sections will begin on September 10-11. Students should indicate their prefer-ences for section time using the student information system. More details will be revealed once the software is complete!
In order to include your name on a section list, we must obtain your permission (on the sectioning form) to reveal on the Web site that you are a student taking Math 23a or E-23a. If you wish to keep this information secret, we will include your name in alphabetical order, but in the form Xxxx Xxxxxx.
Exams: There will be two quizzes and one final exam.
Quiz 1: Wednesday, October 7 (module 1, weeks 1-4) Quiz 2: Wednesday, November 4 (module 2, weeks 5-8) Final Exam: date and time TBA (module 3, weeks 9-12)
Quizzes are held in the Yenching Auditorium, 2 Divinity Avenue. They run from 6 to 9 PM, but you can arrive any time before 7 PM, since 120 minutes should be enough time for the quiz.
Keep these time slots open. Do not, for example, schedule a physics lab or an LS 1a section on Wednesday evenings. If you know that you tend to work slowly, it would also be unwise to schedule another obligation that leaves only part of that time available to you!
Students who have exam accommodations, properly documented by a letter from the Accessible Education Office, may need to take their quizzes in a separate location. Please provide the AEO letters as early in the term as you can, since we may need to reserve one or more extra rooms.
The last day to drop and add courses (like Math 23a and Math 21a) is Monday, October 5. This is before the first quiz. It is important that you be aware of how you are managing the material and performing in the course. It is not a good idea to leave switching out of any course (not just Math 23) until the fifth Mon-day. Decisions of this nature are best dealt with in as timely a manner as possible!! Quizzes will include questions that resemble the ones done in the “early” sec-tions, and each quiz will include two randomly-chosen proofs from among the numbered proofs in the relevant module. There may be other short proofs simi-lar to ones that were done in lecture and problems that are simisimi-lar to homework problems. However if you want quizzes on which you are asked to prove difficult theorems that you have never seen before, you will need to take Math 25a or 55a, not Math 23a.
If you have an unexpected time confilct for one of the quizzes, contact Kate as soon as you know about it, and special arrangements can be made. Distance students will take their quizzes near their home but on the same dates.
The final examination will focus on material from the last five weeks of the course. Local Extension students will take it at the same time and place as under-graduates. The time (9AM or 2PM) will be revealed when the exam schedule is posted late in September. If you have two or even three exams scheduled for that day, don’t worry: that is a problem for the Exams Office, not you, to solve.
Except for the final examination, “local” Extension students can meet all their course obligations after 5:30pm.
“Distance” extension students who do not live near Cambridge and cannot come to Harvard in the evening to hand in homework, attend section and office hours, take quizzes, and present proofs can still participate online in all course activities. Details will be available in a separate document. Since this fully-online
option is an experiment, we plan to restrict it to two sections of 12 students each, with absolute priority given to students who live far from Cambridge.
Textbooks:
Vector Calculus, Linear Algebra, and Differential Forms, Hubbard and Hubbard, fourth edition, Matrix Editions, 2009. Try to get the second printing, which in-cludes a few significant changes to chapters 4 and 6.
This book is in stock at the Coop, or you can order it for $84 plus $10 for priority shipping from the publisher’s Web site at
http://matrixeditions.com/UnifiedApproach4th.html. The Student Solution Manual for the fourth edition, not in stock at the Coop, is also available from that Web site.
We will cover Chapters 1-3 this term, Chapters 4-6 in Math 23b; so this one textbook will last for the entire year.
Ross, Elementary Analysis: The Theory of Calculus, 2nd Edition, 2013. This will be the primary text for the module on single-variable real analysis. It is available electronically through the Harvard library system (use HOLLIS and search for the author and title). If you like to own bound volumes, used copies can be found on amazon.com for as little as $25, but be sure to get the correct edition! Lawvere, Conceptual mathematics: a first introduction to categories, 2nd Edi-tion, 2009.
We will only be using the first chapter, and the book is available for free download through the Harvard library system.
Proofs:
Learning proofs can be fun, and we have put a lot of work into designing an enjoyable way to learn high level and challenging mathematics! Each week’s course materials includes two proofs. Often these proofs appear in the textbook and will also be covered in lecture. They also may appear as quiz questions.
You, as students, will earn points towards your grade by presenting these proofs to teaching staff and to each other without the aid of your course notes. Here is how the system works:
When we first learn a proof in class, only members of the teaching staff are “qual-ified listeners.” Anyone who presents a satisfactory proof to a qual“qual-ified listener also becomes qualified and may listen to proofs by other students. This process of presenting proofs to qualified listeners occurs separately for every proof.
You are expected to present each proof before the date of the quiz on which it might appear; so each proof has a deadline date. Distance students may reference the additional document which details how to go about remotely presenting proofs to classmates and teaching staff.
Each proof is worth 1 point. Here is the grading system:
• Presenting a proof to Paul, Kate, one of the course assistants, or a fellow student who has become a qualified listener: 0.95 points before the deadline, 0.8 points after the deadline. You may only present each proof once.
• Listening to a fellow student’s proof: 0.1 point. Only one student can receive credit for listening to a proof.
• After points have been tallied at the end of the term, members of the course staff may assign the points that they have earned by listening to proofs outside of section to any students that they feel deserve a bit of extra credit. Students who do the proofs early and listen to lots of other students’ proofs can get more than 100%, but there is a cap of 30 points total.You can almost reach this cap by doing each proof before the deadline and listening twice to each proof. Either you do a proof right and get full credit, or you give up and try again later. There is no partial credit. It is OK for the listener to give a couple of small hints.
You may consult the official list of proofs that has the statement of each theorem to be proved, but you may not use notes. That will also be the case when proofs appear on quizzes and on the final exam.
It is your responsibility to use the proof logging software on the course Web site to keep a record of proofs that you present or listen to. You can also use the proof logging software to announce proof parties and to find listeners for your proofs.
Each quiz will include two questions which are proofs chosen at random from the four weeeks of relevant material. The final exam will have three proofs, all from material after the second quiz. Students generally do well on the proof questions.
Useful software: • R and RStudio
This is required only for Extension students who register for graduate credit, but it is an option for everyone. Consider learning R if you
– are interested in computer science and want practice in using software to do things that are more mathematical than can be dealt with in CS 50 or 51.
– are thinking of taking a statistics course, which is likely to use R. – are hoping to get an interesting summer job or summer internship that
uses mathematics or deals with lots of data.
– want to be able to work with large data files in research projects in any field (life sciences, economics and finance, government, etc.)
R is free, open-source software. Instructions for download and installation are on the Web site. You will have the chance to use R at the first section on Thursday, September 10 or Friday, September 11; so install it right away, preferably on a laptop computer that you can bring to section.
On the course Website are a set of R scripts, with accompanying YouTube videos, that explain how to do almost every topic in the course by using R. These scripts are optional for undergraduate, but they will enhance your understanding both of mathematics and of R.
• LaTeX
This is the technology that is used to create all the course handouts. Once you learn how to use it, you can create professional-looking mathematics on your own computer.
The editor that is built into the Canvas course Web site is based on LaTeX. One of the course requirements is to upload four proofs to the course Web site in a medium of your choice. One option is to use LaTeX. Alternatively, you can use the Canvas file editor (LaTeX based), or you can make a YouTube video.
I learned LaTeX without a book or manual by just taking someone else’s files, ripping out all the content, and inserting my own, and so can you. You will need to download freeware MiKTeX version 2.9 (see http://www.miktex.org), which includes an integrated editor named TeXworks.
From http://tug.org/mactex/ you can download a similar package for the Mac OS X.
When in TeXworks, use the Typeset/pdfLaTeX menu item button to create a .pdf file. To learn how to create fractions, sums, vectors, etc., just find an example in the lecture outlines and copy what I did. All the LaTeX source for lecture outlines, assignments, and practice quizzes is on the Web site, so you can find working models for anything that you need to do.
If you create a .pdf file for your homework, please print out the files and hand in the paper at class. An exception can be made if if you are a distance Extension student or for some other good reason you are not in Cambridge on the due date.
The course documents contain examples of diagrams created using TikZ, the built-in graphics editor. It is also easy to include .jpg or .png files in LaTeX. If you want to create diagrams, use Paint or try Inkscape at http://www.inkscape.org, an excellent freeware graphics program. Stu-dents have found numerous other solutions to the problem of creating graph-ics, so just experiment.
If you create a .pdf file for your homework, please print out the files and hand in the paper. By default, undergraduates and “local” Extension students may submit the assignment electronically only if you are out of town on the due date. Individual section instructors may adopt a more liberal poicy about allowing electonic submission. Do not submit .tex files.
Use of R:
You can earn “R bonus points” in three ways:
• By being a member of a group that uploads solutions to section problems that require creation of R scripts. These will be available most, but not all, weeks. (about 10 points)
• By submitting R scripts that solve the optional R homework problems (again available most, but not all, weeks). (about 20 points)
• By doing a term project in R. (about 20 points)
To do the “graduate credit” grade calculation, we wiil add in your R bonus points to the numerator of your score. To the denominator, we will add in 95% of your bonus points or 50% of the possible bonus points, whichever is greater. Earning a lot of R points is essential if you are registered for graduate credit. Oth-erwise,earning more than half the bonus points is certain to raise your percentage score a bit, and it can make a big difference if you have a bad day on a quiz or on the final exam.
Grades: Your course grade will be determined as follows:
• problem sets, 50 points. Your worst score will be converted to a perfect score. • presenting and listening to proofs, 26 points.
• uploading proofs to the Web site, 4 points.
• participation in the “early” sections, based on attendance, preparation, con-tributions to problem solving, and posting solutions to the Web site, 10 points.
• two quizzes, 40 points each.
• final exam, slightly more than 60 points.
• R bonus points, about 50 points in numerator, 25-45 points in denominator. For graduate students, only a “graduate” percentage score, using the R bonus points, will be calculated. For everyone else, we will also calculate an “undergrad-uate” percentage score, ignoring the R bonus points, and we will use the higher of the two percentage scores.
The grading scheme is as follows:
Points Grade 94.0% A 88.0% A-80.0% B+ 75.0% B 69.0% B-63.0% C+ 57.0% C 51.0%
C-If you are conscientious about the homework, proofs, and quizzes, you will end up with a grade between B plus and A, depending on your expertise in taking a fairly long and challenging 3-hour final exam, and you will know that you are thoroughly prepared for more advanced courses. For better or worse, you need to be fast as well as knowledgeable to get an A, but an A- is a reasonable goal even if you make occasional careless errors and are not a speed demon. Extension students who earned a B plus have been successful at getting into PhD programs.
There is no “curve” in this course! You cannot do worse because your classmates do better.
Switching Courses (Harvard College students only):
While transfers among Math 21a, 23a, 25a, and 55a are routine, it is important to note that Math 21a focuses on multivariable calculus, while Math 23a and 25a focus on linear algebra. Math 21b focuses on linear algebra, while Math 23b and 25b focus on multivariable calculus. Math 21a and b are given every semester, while Math 23a and 25a are fall only with 23b and 25b given spring only. Ordinarily there is a small fee if you drop a course after the third Monday of the term, but this is waived in the case of math courses. However, the fifth Monday, October 5, is a firm deadline after which you cannot change courses!
• Math 23a to Math 21a or b
If you decide to transfer out of Math 23a within 3 weeks of the start of the semester, then either Math 21a or 21b is a reasonable choice. If more than 3 weeks have elapsed, Math 21b will be a better place for you to go. You will want to take Math 21a in the spring. You should avoid waiting until the last minute to switch.
Switching to Math 21 at midyear (either to 21b or to 21a) does not make sense except in desperate situations. You will have seen some of the topics in Math 25b, since Math 25a does almost no real analysis. In addition, you will have done about 60% of Math 112, which you are should skip after taking Math 23.
• Math 25a to Math 23a
Math 23a and Math 25a cover similar material during the first three weeks. If you have taken a course in which you learned to multiply matrices and use dot and cross products, you can probably attend only Math 25 lectures for three weeks and still have only a little catching up to do if you add Math 23a during the week of the first quiz. However, if you are trying to decide between 25a and 23a and have not taken a college-level linear algebra course, it might be prudent to attend the lectures in both courses until you make up your mind. Math 23a Weeks 2 and 4 will be new material!
In the case of transfers, graded Math 25a problem sets will be accepted in lieu of missed Math 23a problem sets. It is imperative that you review the problem sets and material that you have missed upon joining the course as soon as possible.
For those who make the decision to change courses at the last minute, there will be special office hours in Science Center 322 on Monday, October 5 from 3 to 4 PM at which study card changes can be approved and arrangements for missed homework and quizzes can be discussed.
Switching from Math 23a to Math 25b at midyear has worked well for a few students over the past several years, although you end up seeing a lot of real analysis twice.
Switching from Math 25a to Math 23b at midyear requires you to teach yourself about multivariable differential calculus and manifolds, but a handful of students do it every year, and it generally works out OK.
Special material for Physics 15b and Physics 153
Math 23b does an excellent treatment of “vector calculus” (div, grad, and curl) and its relation to differential form fields and the exterior derivative. Alas, this material is needed in Physics 15b and Physics 153 before we can reach it in Math 23.
Week 13 covers these topics in a manner that relies only on Math 23a, never mentioning muliple integrals. This will be covered in a special lecture during reading period, and there will be a optional ungraded problem set. If you choose to do this topic, which physics students last year said was extremely useful, there will be one question about it on the final exam, which you can use to replace your lowest score on one of the other questions.
If you are not taking Physics 15b or Physics 153, just wait to see this material in Math 23b.
YouTube videos
These were made as part of a rather unsuccessful pedagogical experiment last year. They are quite good, but you will need some extra time to watch them.
• The Lecture Preview Videos were made by Kate. They cover the so-called Executive Summaries in the weekly course materials, which go over all the course materials, but without proofs or detailed examples.
If you watch these videos (it takes about an hour per week) you will be very well prepared for lecture, and even the most difficult material will make sense on a first hearing.
Last year’s experiment was unsuccessful because we assumed in lecture that everyone had watched these videos, when in fact only half the class did so. Those who did not watch them complained, correctly, that the lectures skipped over basic material in getting to proofs and examples. This year’s lectures will be self-contained, so the preview videos are not required viewing. • The R script videos were made by Paul. They provide a line-by-line
expla-nation of the R scripts that accompany each week’s materials.
Last year’s experiment was unsuccessful because going over these scripts in class was not a good use of lecture time. If you are doing the “graduate” option, these scripts are pretty much required viewing, although the scripts are so thoroughly commented that just working through them on your own is perhaps a viable alternative.
If you are doing just the “undergraduate” option, you can ignore the R scripts completely.
Homework: Homework (typically 8 problems) will be assigned weekly. The assignment will be included in the same online document as the lecture notes and section problems.
Assignments are due on Wednesdays by 10:00 AM. There will be a locked box on the second floor, near Room 209, with your “late” section instructor’s name. At 10 AM Kate will place a sheet of colored paper in each box, and anything above that paper will be late! Please include your name, the assignment number, and your CA’s name on your assignment.
Each week’s assignment will include a couple of optional problems whose so-lutions require R scripts. These scripts should be uploaded electronically to the dropbox on the Web site for that week. Please include your name as a comment in the script and also in the file name.
The course assistant who leads your “late” section should return your corrected homework to you at the section after the due date. If you are not receiving graded homework on schedule, send email to [email protected] and the problem will be dealt with.
Homework that is handed in after 10AM on the Wednesday when it is due will not be graded. If it arrives before the end of Reading Period and looks fairly complete, you will get a grade of 50% for it.
It is a violation of Federal privacy law for us to return graded homework by placing it in a publicly accessible location like an instructor’s mailbox. You will have to collect your graded homework from your section instructor in person.
Collaboration and Academic Integrity policy:
You are encouraged to discuss the course with other students and with the course staff, but you must always write your homework solutions out yourself in your own words. You must write the names of those you’ve collaborated with at the top of your assignment.
If you collaborate with classmates to solve problems that call for R scripts, create your own file after your study group has figured out how to do it.
Proofs that you submit to the course Web site must be done without consulting files that other students have posted!
If you have the opportunity to see a complete solution to an assigned problem, please refrain from doing so. If you cannot resist the temptation, you must cite the source, even if all that you do is check that your own answer is correct.
You are forbidden to upload solutions to homework problems, whether your own or ones that are posted on the course Web site, to any publicly available location on the Internet.
Anything that you learn from lecture, from the textbook, or from working homework problems can be regarded as “general knowledge” for purposes of this course, and the source need not be cited. Anything learned in prerequisite courses falls into the same category. Do not assume that other courses use some an ex-pansive definition of “general knowledge”!
Tutoring: Several excellent students from previous years, qualified to be course assistants but too busy, are registered with the Bureau of Study Counsel as tutors. If you find yourself getting into difficulties, immediately contact the BSC and get teamed up with one of them.
You will have to contact the BSC directly to arrange for a tutor, since privacy law forbids anyone on the Math 23 staff to know who is receiving tutoring. A website with more information can be found at www.bsc.harvard.edu.
Week-by-week Schedule:
Month Date Topic
Fortnight 1 September 3-11 Fields, vectors and matrices
Week 2 September15-18 Dot and cross products; Euclidean geometry of Rn
Week 3 September 22-25 Row reduction, independence, basis Week 4 Sept. 29 - Oct. 2 Eigenvectors and eigenvalues
Week 5 October 6-9 Number systems and sequences October 7 QUIZ 1 on weeks 1-4
Week 6 October 13-16 Series, convergence tests, power series Week 7 October 20-23 Limits and continuity of functions
Week 8 October 27-30 Derivatives, inverse functions, Taylor series
Week 9 November 3-6 Topology, sequences in Rn, linear differential equations October 29 QUIZ 2 on weeks 5-8
Week 10 November 10-13 Limits and continuity in Rn; partial and directional derivatives
Week 11 November 17-20 Differentiability, Newton’s method, inverse functions Fortnight 12 Nov. 24-Dec. 3 Manifolds, critical points, Lagrange multipliers
November 26 Thansksgiving
Half-week 13 December 8 Calculus on parametrized curves; div, grad, and curl December ? FINAL EXAM on weeks 9-12
This schedule covers all the math that is needed for Physics 15a, 16, and 15b with the sole exception of surface integrals, which will be done in the spring. The real analysis in Math 23a alone will be sufficient for most PhD programs in economics, though the most prestigious programs will want to see Math 23b also. All the mathematics that is used in Economics 1011a will be covered by the end of the term. The coverage of proofs is complete enough to permit prospective Computer Science concentrators to skip CS 20.
Abstract vector spaces and multiple integration, topics of great importance to prospective math concentrators, have all been moved to Math 23b.
MATHEMATICS 23a/E-23a, Fall 2016 Linear Algebra and Real Analysis I
Module #1, Week 1 (Fields, Vectors, and Matrices) Authors: Paul Bamberg and Kate Penner
R scripts by Paul Bamberg
Last modified: June 13, 2015 by Paul Bamberg
Reading
• Hubbard, Sections 0.1 through 0.4 • Hubbard, Sections 1.1, 1.2, and 1.3
• Lawvere and Schanuel, Conceptual Mathematics
Search the Internet for ”Harvard HOLLIS” and type ”Conceptual Mathe-matics” into the Search box.
Choose View Online. You will have to log in with your Harvard PIN. At a minimum, read the following:
Article I (Sets, maps, composition – definition of a category) Session 2
This is very easy reading.
Proofs to present in section or to a classmate who has done them. • 1.1 Suppose that a and b are two elements of a field F . Using only the
axioms for a field, prove the following: – If ab = 0, then either a or b must be 0. – The additive inverse of a is unique.
• 1.2(Generalization of Hubbard, proposition 1.2.9) A is an n × m matrix. The entry in row i, column j is ai,j
B is an m × p matrix. C is an p × q matrix.
The entries in these matrices are all from the same field F . Using summa-tion notasumma-tion, prove that matrix multiplicasumma-tion is associative:
that (AB)C = A(BC). Include a diagram showing how you would lay out the calculation in each case so the intermediate results do not have to be recopied.
– a. Suppose that the matrix [T ] is invertible. Prove that the linear transformation T is one-to-one and onto (injective and surjective), hence invertible.
– b. Suppose that linear transformation T is invertible. Prove that its inverse S is linear and that the matrix of S is [S] = [T ]−1
Note: Use * to denote matrix multiplication and ◦ to denote composition of linear transformations. You may take it as already proved that matrix multiplication represents composition of linear transformations. Do not assume that m = n. That is true, but we are far from being able to prove it, and you do not need it for the proof.
R Scripts
• Script 1.1A-Finite Fields.R
Topic 1 - Why the real numbers form a field
Topic 2 - Making a finite field, with only five elements Topic 3 - A useful rule for finding multiplicative inverses • Script 1.1B-PointsVectors.R
Topic 1 - Addition of vectors in R2
Topic 2 - A diagram to illustrate the point-vector relationship Topic 3 - Subtraction and scalar multiplication
• Script 1.1C-Matrices.R
Topic 1 - Matrices and Matrix Operations in R Topic 2 - Solving equations using matrices Topic 3 - Linear functions and matrices Topic 4 - Matrices that are not square Topic 5 - Properties of the determinant • Script 1.1D-MarkovMatrix
Topic 1 - A game of volleyball
Topic 2 - traveling around on ferryboats • Script 1.1L-LinearMystery
1
Executive Summary
• Quantifiers and Negation Rules
The “universal quantifier” ∀ is read “for all.”
The “existential quantifier” exists is read “there exists.” It is usually followed by “s.t,” a standard abbreviation for “such that.”
The negation of “∀x, P (x) is true” is “∃x, P (x) is not true.” The negation of “∃x, P (x) is true” is “∀x, P (x) is not true.”
The negation of “P and Q are true” is “either P or Q is not true.” The negation of “either P or Q is true” is “both P and Q are not true.”
• Functions
A function f needs two sets: its domain X and its codomain Y .
f is a rule that, to any element x ∈ X, assigns a specific element y ∈ Y . We write y = f (x)
f must assign a value to every x ∈ X, but not every y ∈ Y must be of the form f (x). The subset of the codomain consisting of elements that are of the form y = f (x) is called the image of f . If the image of f is all of the codomain Y , f is called surjective or onto
f need not assign different of elements of Y to different elements of X. If x1 6= x2 =⇒ f (x1) 6= f (x2), f is called injective or one-to-one
If f is both surjective and injective, it is bijective and has an inverse f−1.
• Categories
A category C has objects (which might be sets) and arrows (which might be functions)
An arrow f must have a specific domain objectX and a specific codomain object Y ; we write f : X → Y or X −→ Y .f
If arrows f : X → Y and g : Y → Z are in the category, then the composi-tion arrow f ◦ g : X → Z is in the category.
For any object X there is an identity arrow IX : x → X
Given f : X → Y , f ◦ IX = f and IY ◦ f = f .
Associative law: given X −→ Yf −→ Zg −→ W , h ◦ (g ◦ f ) = (h ◦ g) ◦ fh
Given an arrow f : X → Y , an arrow g : Y → X such that g ◦ f = IX is
called a retraction.
Given an arrow f : X → Y , an arrow g : Y → X such that f ◦ g = IY is
called a section.
1.1
Fields and Field Axioms
A field F is a set of elements for which the familiar operations of addition and multiplication are defined and behave in the usual way. Here is a set of axioms for a field. You can use them to prove theorems that are true for any field.
1. Addition is commutative: a + b = b + a.
2. Addition is associative: (a + b) + c = a + (b + c).
3. Additive identity: ∃0 such that ∀a ∈ F, 0 + a = a + 0 = a.
4. Additive inverse: ∀a ∈ F, ∃ − a such that −a + a = a + (−a) = 0. 5. Multiplication is associative: (ab)c = a(bc).
6. Multiplication is commutative: ab = ba.
7. Multiplicative identity: ∃1 such that ∀a ∈ F, 1a = a.
8. Multiplicative inverse: ∀a ∈ F − {0}, ∃a−1 such that a−1a = 1. 9. Distributive law: a(b + c) = ab + ac.
Examples of fields include: The rational numbers Q. The real numbers R. The complex numbers C.
The finite field Zp, constructed for any prime number p as follows:
• Break up the set of integers into p subsets. Each subset is named after the remainder when any of its elements is divided by p.
[a]p = {m|m = np + a, n ∈ Z}
Notice that [a + kp]p = [a]p for any k. There are only p sets, but each has
many alternate names. These p infinite sets are the elements of the field Zp.
• Define addition by [a]p+ [b]p = [a + b]p. Here a and b can be any names for
the subsets, because the answer is independent of the choice of name. The rule is “Add a and b, then divide by p and keep the remainder.”
• Define multiplication by [a]p[b]p = [ab]p. Again a and b can be any names
for the subsets, because the answer is independent of the choice of name. The rule is “Multiply a and b, then divide by p and keep the remainder.”
1.2
Points and Vectors
Fn denotes the set of ordered lists of n elements from a field F . Usually the field is R, but it could be the field of complex numbers C or a finite field like Z5.
A given element of Fn can be regarded either as a point, which represents
“position data,” or as a vector, which represents “incremental data.”
If an element of Fnis a point, we represent it by a bold letter like p and write it as a column of elements enclosed in parentheses.
p = 1.1 −3.8 2.3
If an element of Fn is a vector, we represent it by a bold letter with an arrow like ~v and write it as a column of elements enclosed in square brackets.
~ v = −0.2 1.3 2.2
To add a vector to a point, we add the components in identical positions together. The result is a point: q = p + ~v. Geometrically we represent this by anchoring the vector at the initial point p. The location of the arrowhead of the vector is the point q that represents our sum.
p
q ~
v
To add a vector to a vector, we again add component by component. The result is a vector. Geometrically, the vector created by beginning at the initial point of the first vector and ending at the arrowhead of the second vector is the represents our sum.
~ v
~ w ~v + ~w
To form a scalar multiple of a vector, we multiply each component by the scalar. In Rn, the geometrical effect is to multiply the length of the vector by the scalar. If the scalar is a negative number, we switch the position of the arrow to the other end of the vector.
1.3
Standard basis vectors
The standard basis vector ~ek has a 1 as its kth component, and all its other
components are 0. Since the additive identity 0 and the multiplicative identity 1 must be present an any field, there will always be n standard basis vectors in Fn. Geometrically, the standard basis vectors in R2 are usually associated with ”one unit east” and ”one unit north” respectively.
~e1
~e2
1.4
Matrices and linear transformations
An m × n matrix over a field F has m rows and n columns.
Matrices represent linear functions, also known as linear transformations: A function g : Fn → Fm is called linear if
g(a~v + b ~w) = ag(~v) + bg( ~w).
For a linear function g, if we know the value of g(~ei) for each standard basis
vector ~ei, the value of g(~v) for any vector v follows by linearity:
g(v1~e1+ v2~e2+ · · · + vn~en) = v1g(~e1) + v2g(~e2) + · · · + vng(~en)
The matrix G that represents the linear function g is formed by using g(~ek)
as the kth column. Then, if gi,j denotes the entry in the ith row and jth column
of matrix G, the function value ~w = g(~v) can be computed by the rule
wi = n X j=1 gi,jvj
1.5
Matrix multiplication
If m × n matrix G represents linear function g : Fn → Fm and n × p matrix H
represents linear function h : Fp → Fn, then the matrix product GH is defined
so that it represents their composition: the linear function g ◦ h : Fp → Fm.
Start with standard basis vector ~ej. Function h converts this to the jth
column ~hj of matrix H. Then function g converts this column to g(~hj), which
must therefore be the jth column of matrix GH.
The rule for forming the product GH can be stated in terms of the rule for a matrix acting on a vector: to form GH, just multiply G by each column of H in turn, and put the results side by side to create the matrix GH. If C = GH,
1.6
Examples of matrix multiplication
B 0 1 2 −1 −2 0 A 2 1 0 1 −1 −2 A2 1 0 1 −1 −2 2 1 −6 2 AB B 0 1 2 −1 −2 0 1 −1 2 3 3 −2 −4 −2 0 BA The number of columns in the first factor must equal the number of rows in the second factor.1.7
Function inverses
A function f : X → Y is invertible if it has the following two properties: • It is injective (one-to-one): if f (x1) = f (x2) , then x1 = x2.
• It is surjective (onto): ∀y ∈ Y, ∃x ∈ X such that f (x) = y.
The inverse function g = f−1 has the property that if f (x) = y then g(y) = x. So g(f (x)) = x and f (g(y)) = y. Both f ◦ g and g ◦ f are the identity function.
1.8
The determinant of a 2 × 2 matrix
For matrix A = a b c d
, det A = ad − bc. If you fix one column, it is a linear function of the other column, and it changes sign if you swap the two columns.
1.9
Matrix inverses
A non-square m × n matrix A can have a “one-sided inverse.”
If m > n, then A takes a vector in Rn and produces a longer vector in Rm.
In general, there will be many matrices B that can recover the original vector in Rn, so that BA = In. In this case there is no right inverse.
If m < n, then A takes a vector in Rn and produces a shorter vector in Rm.
In general, there will be no left inverse matrix B that can recover the original vector in Rn, but there may be many different right inverses for which AB = I
m.
For a square matrix, it is possible for both a right inverse B and a left inverse C to exist. In this case, we can prove that B and C are equal and they are unique. We can say that “an inverse” A−1 exists, and it represents the inverse of the linear function represented by matrix A.
You can find the inverse of a 2 × 2 matrix A whose determinant is not zero by using the formula
1.10
Matrix transposes
The transpose of a given matrix A is written AT. The two are closely related. The rows of A are the columns of AT and the columns of A are the rows of AT.
A =a b c d , AT =a c b d
The transpose of a matrix product is the product of the transposes, but in the opposite order:
(AB)T = BTAT A similar rule holds for matrix inverses:
(AB)−1 = B−1A−1
1.11
Applications of matrix multiplication
In these examples, the “sum of products” rule for matrix multiilpication arises naturally, and so it is efficient to use matrix techniques.
• Counting paths: Suppose we have four islands connected by ferry routes: 1
2
3
4
The entry in row i, column j of the matrix A = 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 0 shows how
many ways there are to reach island i by a single ferry ride, starting from island j. The entry in row i, column j of the matrix An shows how many ways there are to reach island i by a sequence of n ferry rides, sarting from island j.
• Markov processes: A game of beach volleyball has two “states”: in state 1, team 1 is serving, in state 2, team 2 is serving. With each point that is played there is a “state transition” governed by probabilities: for exam-ple, from state 1, there is a probability of 0.8 of remaining in state 1, a
2
Lecture Outline
1. Quantifiers and negation
Especially when you are explaining a proof to someone, it saves some writing to use the symbols ∃ (there exists) and ∀ (for all).
Be careful when negating these.
The negation of “∀x, P (x) is true” is “∃x, P (x) is not true.” The negation of “∃x, P (x) is true” is “∀x, P (x) is not true.” When negating a statement, also bear in mind that
The negation of “P and Q are true” is “either P or Q is not true.” The negation of “either P or Q is true” is “both P and Q are not true.” For practice, let’s negate the following statements (which may or may not be true!)
• There exists an even prime number. Negation:
• All 11-legged alligators are orange with blue spots. (Hubbard, page 5) Negation:
• The function f (x) is continuous on the open interval (0,1), which means that ∀x ∈ (0, 1), ∀ > 0, ∃δ > 0 such that ∀y ∈ (0, 1),
|y − x| < δ implies |f (y) − f (x)| < .
2. Set notation
Here are the standard set-theoretic symbols: • ∈ (is an element of)
• {a|p(a)} (set of a for which p(a) is true) • ⊂ (is a subset of)
• ∩ (intersection) • ∪ (union)
• × (Cartesian product) • - or \ (set difference)
Using the integers Z and the real numbers R, let’s construct some sets. In each case there is one way to describe the set using a restriction and another more constructive way to describe the set.
• The set of real numbers whose cube is greater than 8 in magnitude. Restrictive:
Constructive:
• The set of coordinate pairs for points on the circle of radius 2 centered at the origin (an example of a “smooth manifold”).
Restrictive:
3. Function terminology:
Here are some terms that should be familiar from your study of precalculus and calculus:
Example a
Example b
Example c
domain
codomain
image
one-to-one = injective
onto = surjective
invertible = bijective
Using the sets X = {1, 2} and Y = {A, B, C}, draw diagrams to illustrate the following functions, and fill in the table to show how the terms apply to them:
• f : X → Y, f (1) = A, f (2) = B.
• g : Y → X, g(A) = 1, g(B) = 2, g(C) = 1.
Here are those function words again, with two additions: • domain
• natural domain (often deduced from a formula) • codomain • image • one-to-one = injective • onto = surjective • invertible = bijective • inverse image = {x|f (x) ∈ A}
Here are functions from R to R, defined by formulas. • f1(x) = x2
• f2(x) = x3
• f3(x) = log x(natural logarithm)
• f4(x) = ex
• Find one that is not injective (not one-to-one)
• For f1, what is the inverse image of (1, 4)?
• Which function is invertible as a function from R to R?
• What is the natural domain of f3?
• What is the image of f4?
• Specify domain and codomain so that f3 and f4 are inverses of one
4. Composition of functions
Sometimes people find that a statement is hard to prove because it is so obvious. An example is the associativity of function composition, which will turn out to be crucial for linear algebra.
Prove that (f ◦ g) ◦ h = f ◦ (g ◦ h). Hint: Two functions f1 and f2 are equal
if they have the same domain X and, ∀x ∈ X, f1(x) = f2(x).
Consider the set of men who have exactly one brother and least one son. h(x) = “father of x”, g(x) = “brother of x”, f (x) = “oldest son of x”
• f ◦ g is called
• (f ◦ g) ◦ h is
• g ◦ h is called
• f ◦ (g ◦ h) is
• Simpler name for both (f ◦ g) ◦ h and f ◦ (g ◦ h)
Consider the real-valued functions g(x) = ex, h(x) = 3 log x, f (x) = x2
• f ◦ g has the formula
• (f ◦ g) ◦ h has the formula
• g ◦ h has the formula
• f ◦ (g ◦ h) has the formula
5. Finite sets and functions form the simplest example of a category • The objects of the category are finite sets.
• The arrows of the category are functions from one finite set to another. The definition of a function involves quantifiers.
Requirements for a function f : X → Y ∀x ∈ X, ∃!y ∈ Y such that f (x) = y What is wrong with the following?
X Y
What is wrong with the following?
X Y
• If arrows f : X → Y and g : Y → Z are in the category, then the composition arrow f ◦ g : X → Z is in the category.
• For any object X there is an identity arrow IX : x → X
• Given f : X → Y , f ◦ IX = f and IY ◦ f = f
• Composition of arrows is associative:
Given X −→ Yf −→ Zg −→ W , h ◦ (g ◦ f ) = (h ◦ g) ◦ fh
The objects do not have to be sets and the arrows do not have to be functions. For example, the objects could be courses, and an arrow from course X to course Y could mean ”if you have taken course X, you will probably do better in course Y as a result.” Check that the identity and composition rules are satisfied.
6. Invertible functions - an example of invertible arrows
First consider the category of finite sets and functions between them. The term “inverse” is used only for a “two-sided inverse.” Given f : X → Y , an inverse f−1 : Y → X must have the properties
f−1◦ f = IX and f ◦ f−1 = IY
Prove that the inverse is unique. This proof uses only things that are true in any category, so it is valid in any category!
This function is not invertible because it is not injective, but it is surjective.
X Y
However, it has a “preinverse” (my terminology – the official word is “sec-tion.”) Starting at an element of Y , choose any element of X from which there is an arrow to that element. Call that function g. Then f ◦ g = IY
but g ◦ f 6= IX. Furthermore, g is not unique.
Prove the cancellation law that if f has a section and h ◦ f = k ◦ h, then h = k (another proof that is valid in any category!)
This function f is not invertible because it is not surjective, but it is injec-tive.
7. Fields
Loosely speaking, a field F is a set of elements for which the familiar oper-ations of arithmetic are defined and behave in the usual way. Here is a set of axioms for a field. You can use them to prove theorems that are true for any field.
(a) Addition is commutative: a + b = b + a.
(b) Addition is associative: (a + b) + c = a + (b + c).
(c) Additive identity: ∃0 such that ∀a ∈ F, 0 + a = a + 0 = a.
(d) Additive inverse: ∀a ∈ F, ∃ − a such that −a + a = a + (−a) = 0. (e) Multiplication is associative: (ab)c = a(bc).
(f) Multiplication is commutative: ab = ba.
(g) Multiplicative identity: ∃1 such that ∀a ∈ F, 1a = a.
(h) Multiplicative inverse: ∀a ∈ F − {0}, ∃a−1 such that a−1a = 1. (i) Distributive law: a(b + c) = ab + ac.
This set of axioms for a field includes properties (such as the commutativity of addition) that can be proved as theorems by using the other axioms. It therefore does not qualify as an “independent” set, but there is no general requirement that axioms be independent.
Some well-known laws of arithmetic are omitted from the list of axioms because they are easily proved as theorems. The most obvious omission is ∀a ∈ F, 0a = 0.
Here is the proof. What axiom justifies each step?
• 0 + 0 = 0 so (0 + 0)a = 0a.
• 0a + 0a = 0a.
• (0a + 0a) + (−0a) = 0a + (−0a).
• 0a + (0a + (−0a)) = 0a + (−0a).
• 0a + 0 = 0.
8. Finite fields
Computing with real numbers by hand can be a pain, and most of linear algebra works for an arbitrary field, not just for the real and complex num-bers. Alas, the integers do not form a field because in general there is no multiplicative inverse. Here is a simple way to make from the integers a finite field in which messy fractions cannot arise.
• Choose a prime number p.
• Break up the set of integers into p subsets. Each subset is named after the remainder when any of its elements is divided by p.
[0]p = {m|m = np, n ∈ Z}
[1]p = {m|m = np + 1, n ∈ Z}
[a]p = {m|m = np + a, n ∈ Z}
Notice that [a + kp]p = [a]p for any k. There are only p sets, but each
has many alternate names.
These p infinite sets are the elements of the field Zp.
• Define addition by [a]p + [b]p = [a + b]p. Here a and b can be any
names for the subsets, because the answer is independent of the choice of name. The rule is “Add a and b, then divide by p and keep the remainder.”
• What is the simplest name for [5]7+ [4]7?
• What is the simplest name for the additive inverse of [3]7?
• Define multiplication by [a]p[b]p = [ab]p. Again a and b can be any
names for the subsets, because the answer is independent of the choice of name. The rule is “Multiply a and b, then divide by p and keep the remainder.”
• What is the simplest name for [5]7[4]7?
9. Rational numbers
The rational numbers Q form a field. You learned how to add and multiply them years ago! The multiplicative inverse of ab is ab as long as a 6= 0. The rational numbers are not a “big enough” field for doing Euclidean geometry or calculus. Here are some irrational quantities:
• √2 • π.
• most values of trig functions, exponentials, or logarithms. • coordinates of most intersections of two circles.
10. Real numbers
The real numbers R constitute a field that is large enough so that any characterization of a number in terms of an infinite sequence of real numbers still leads to a real number.
A positive real number is an expression like 3.141592... where there is no limit to the number of decimal places that can be provided if requested. To get a negative number, put a minus sign in front. This is Hubbard’s definition.
An equivalent viewpoint is that a positive real number is the sum of an integer and an infinite series of the form
∞ X i=1 ai( 1 10) i
where each ai is one of the decimal digits 0...9.
Write the first three terms of an infinite series that converges to π.
The rational numbers and the real numbers are both “ordered fields.” This means that there is a subset of positive elements that is closed under both addition and multiplication. No finite field is ordered.
In Z5, you can name the elements [0], [1], [2], [−2], [−1], and try to call the
elements [1] and [2] “positive.” Why does this attempt to make an ordered field fail?
11. Proof 1.1 - two theorems that are valid in any field
(a) Using nothing but the field axioms, prove that if ab = 0, then either a or b must be 0.
(b) Using nothing but the field axioms, prove that the additive inverse of an element a is unique. (Standard strategy for uniqueness proofs: assume that there are two different inverses b and c, and prove that b = c.
12. Lists of field elements as points and vectors:
Fn denotes the set of ordered lists of n elements from a field F . Usually the field is R, but it could be the field of complex numbers C or a finite field like Z5.
An element of Fn can be regarded either as a point, which represents
“po-sition data,” or as a vector, which represents “incremental data.” Beware: many textbooks ignore this distinction!
If an element of Fn is a point, we represent it by a bold letter like p and write it as a column of elements enclosed in parentheses.
p = 1.1 −3.8 2.3 ,
If an element of Fn is a vector, we represent it by by a bold letter with an arrow like ~v and write it as a column of elements enclosed in square brackets. ~ v = −0.2 1.3 2.2
13. Relation between points and vectors, inspired by geometry:
• Add vector ~v component by component to point A to get point B. • Subtract point A component by component from point B to get vector
~ v.
• Vector addition: if adding ~v to point A gives point B and adding ~w to point B gives point C, then adding ~v + ~w to point A gives point C.
• A vector in Fn can be multiplied by any element of F to get another
vector.
Draw a diagram to illustrate these operations without use of coordinates, as is typically done in a physics course.
14. Examples from coordinate geometry Here are two points in the plane.
p = 1.4 −3.8 , q = 2.4 −4.8
Here are two vectors.
~ v =−0.2 1.3 , ~w = 0.6 −0.2 • What is q − p? • What is p + ~v? • What is ~v − 1.5 ~w? • What, if anything, is p + q?
15. Subsets of Fn
A subset of Fn can be finite, countably infinite, or uncountably infinite. The concept is especially useful when the elements of Fn are points, but it
is valid also for vectors. Examples:
(a) In Z2
3, consider the set {
0 1 ,1 2 ,2 0 }.
This will turn out (outline 7) to be a line in “the small affine faculty senate.” Write it in the form {p + t~v|t ∈ Z3}.
(b) In R2, consider the set of points whose coordinates are both positive integers. Is it finite, countably infinite, or uncountably infinite?
(c) In R2, consider the set of points on the unit circle, a “one-dimensional
manifold.” Is it finite, countably infinite, or uncountably infinite?
(d) In R2, draw a diagram that might represent the set of points x
y
, where x is family income and y is family net worth, for which a family qualifies for free tuition.
16. Subspaces of Fn
A subspace is defined only when the elements of Fn are vectors. It must be closed under vector addition and scalar multiplication. The second re-quirement means that the zero vector must be in the subspace. The empty set ∅ is not a subspace!
Geometrically, a subspace corresponds to a “flat subset” (line, plane, etc.) that includes the origin.
For R3 there are four types of subspace. What is the geometric interpreta-tion of each?
• 0-dimensional: the set { 0 0 0 } • 1-dimensional: {t~u|t ∈ R} Exception: 0-dimensional if • 2-dimensional: {s~u + t~v|s, t ∈ R} Exception: 1-dimensional if • 3-dimensional: {r~u + s~v + t ~w|r, s, t ∈ R} Exceptions: 2-dimensional if 1-dimensional if
A special type of subset is obtained by adding all the vectors in a subspace to a fixed point. It is in general not a subspace, but it has special properties. Lines and planes that do not contain the origin fall into this category. We call such a subset an “affine subset.” This terminology is not standard:
17. Standard basis vectors:
These are useful when we want to think of Fn more abstractly.
The standard basis vector ~eihas a 1 in position i, a 0 everywhere else. Since
0 and 1 are in every field, these vectors are defined for any F .
The nice thing about standard basis vectors is that in Fn, any vector can
be represented uniquely in the form
n
X
i=1
xie~i
This will turn out to be true also in an abstract n-dimensional vector space, but in that case there will be no “standard” basis.
18. Another meaning for “field”
Physicists long ago started using the term “field” to mean “a function that assigns a vector to every point.” Examples are the gravitational field, electric field, and magnetic field.
Another example: in a smoothly flowing stream or in a blood vessel, there is a function that assigns to each point the velocity vector of the fluid at that point: a “velocity field.”
If x1 x2
is the point whose coordinates are the interest rate x1 and the
unemployment rate x2, then the Fed chairman probably has in mind the
function that assigns to this point a vector: the expected change in these quantities over the next month.
A function ~F x1 x2
that assigns to this point a vector of rates of change: dx1 dt dx2 dt = ~F x1 x2
specifies a linear differential equation involving two variables. In November you will learn to solve such equations by matrix methods.
Here is a formula for a vector field from Hubbard, exercise 1.1.6 (b). Plot it. ~ F x y =x 0 .
Here are formulas for vector fields from Hubbard, exercise 1.1.6, (c) and (e). Plot them. If you did Physics C Advanced Placement E&M, they may look familiar. ~ F x y =x y , F~x y =−y x 19. Matrices
An m × n matrix over a field F is a rectangular array of elements of F with m rows and n columns. Watch the convention: the height is specified first! As a mathematical object, any matrix can be multiplied by any element of F . This could be meaningless in the context of an application. Suppose you run a small hospital that has two rooms with three patients in each. Then
98.6 102.4 99.7 103.2 98.3 99.6
is a perfectly reasonable way to keep track of the body temperatures of the patients, but multiplying it by 2.7 seems unreasonable. This matrix, viewed as an element of R6, is a point, not a vector, but we always use braces for
matrices.
Matrices with the same size and shape can be added component by com-ponent. What would you get if you add
0.2 −1.4 0.0 0.6 −0.9 2.35
20. Matrix multiplication
Matrix multiplication is nicely explained on pp. 43-46 of Hubbard. To illustrate the rule, we will take
A =2 1 0 1 −1 −2 , B = 0 1 2 −1 −2 0 • Compute AB. 0 1 2 −1 −2 0 2 1 0 1 −1 −2 • Compute BA. 2 1 0 1 −1 −2 0 1 2 −1 −2 0
In a set of n×n square matrices, addition and multiplication of matrices are always defined. Multiplication is distributive with respect to addition, too. But because matrix multiplication is noncommutative, the n × n matrices do not form a field if n > 1. (They are said to form a ring.) Let
A =1 1 1 0 B =0 1 2 1 Find AB. 0 1 2 1 1 1 1 0 Find BA. 1 1 1 0
21. Matrices as functions:
Since a column vector is also an n × 1 matrix, we can multiply an m × n matrix by a vector in Fn to get a vector in Fm. The product A~e
i is the
ith column of A. This is usually the best way to think of a matrix A as representing a linear function f : the ith column of A is f (~ei).
Example: Suppose that f (1 0 ) = 1 4 , f (0 1 ) = 2 3 . What matrix represents f ?
Since A(xie~i+ xje~j) is the sum of xi times column i and xj times column
j, we see that
f (xie~i+ xje~j) = xif (~ei) + xjf (~ej)
This is a requirement if f is to be a linear function. Use matrix multiplication to calculate f ( 2
−1
).
The rule for forming the product AB can be stated in terms of the rule for a matrix acting on a vector: to form AB, just let A act on each column of B in turn, and put the results side by side to create the matrix AB. What function does the matrix product AB represent? Consider (AB)~ei.
This is the ith column of the matrix AB, and it is also the result of letting B act on ~ei, then letting A act on the result. So for any standard basis
vector, the matrix AB represents the composition A ◦ B of the functions represented by B and by A.
What about the matrices (AB)C and A(BC)? These represent the compo-sition of three functions: say (f ◦ g) ◦ h and f ◦ (g ◦ h). But we already know that composition of functions is associative. So we have proved, without any messy algebra, that multiplication of matrices is associative also.
22. Proving associativity by brute force (proof 1.2) A is an n × m matrix.
B is an m × p matrix. C is an p × q matrix.
What is the shape of the matrix ABC?
Show how you would lay out the calculation of (AB)C.
If ai,j represents the entry in the ith row, jth column of A, then
(AB)i,k = m X j=1 ai,jbj,k ((AB)C)i,q= p X k=1 (AB)i,kck,q = m X j=1 p X k=1 (ai,jbj,k)ck,q
Show how you would lay out the calculation of A(BC).
(BC)
j,q=
(A(BC))
i,q=
On what basis can you now conclude that matrix multiplication is associa-tive for matrices over any field F ?
23. Identity matrix:
It must be square, and the ith column is the ith basis vector. For example,
I3 = 1 0 0 0 1 0 0 0 1 24. Matrices as the arrows for a category C
Choose a field F , perhaps the real numbers R. • An object of C is a vector space Fn.
• An arrow of C is an n × m matrix A, with domain Fm and codomain
Fn.
• Given Fp −→ FB m −→ FA n the composition of arrows A and B is the
matrix product AB. Show that the “shape” of the matrices is right for multiplication.
• The identity arrow for object Fn is the n × n identity matrix.
Now we just have to check the two rules that must hold in any category:
• The associative law for composition of arrows holds because, as we just proved, matrix multiplication is associative.
• Verify the two identity rules for the case where A =2 3 4 1 2 3
25. Matrix inverses:
Consider first the case of a non-square m × n matrix A.
If m > n, then A takes a vector in Rn and produces a longer vector in
Rm. In general, there will be many matrices B that can recover the original vector in Rn. In the lingo of categories, such a matrix B is a retraction. Here is a matrix that converts a 2-component vector (price of silver and price of gold) into a three-component vector that specifies the price of alloys containing 25%, 50%, and 75% gold respectively. Calculate ~v = A4
8 . A = .75 .25 .5 .5 .25 .75 , ~v = A 4 8 =
By elementary algebra you can reconstruct the price of silver and of gold from the price of any two of the alloys, so it is no surprise to find two different left inverses. Apply each of the following to ~v.
B1 = 2 −1 0 −2 3 0 , B1~v = B2 = 0 3 −2 0 −1 2 , B2~v =
However, in this case there is no right inverse.
If m < n, then A takes a vector in Rn and produces a shorter vector in
Rm. In general, there will be no left inverse matrix B that can recover the original vector in Rn, but there may be many different right inverses. Let
A =1 −1 and find two different right inverses. In the lingo of categories, such a matrix A is a section.
26. Inverting square matrices
For a square matrix, the interesting case is where both a right inverse B and a left inverse C exist. In this case, B and C are equal and they are unique. We can say that “an inverse” A−1 exists.
Proof of both uniqueness and equality:
To prove uniqueness of the left inverse matrix, assume that matrix A has two different left inverses C and C0 and a right inverse B:
C0A = CA = I C0(AB) = C(AB) = IB
C0I = CI = B C0 = C = B
In general, inversion of matrices is best done by “row reduction,” discussed in Chapter 2 of Hubbard. For 2 × 2 matrices there is a simple formula that is worth memorizing: If A =a b c d then A−1 = 1 ad − bc d −b −c a
If ad − bc = 0 then no inverse exists. Write down the inverse of 3 1
4 2
The matrix inversion recipe works in any field: try inverting A =3 1
4 2
where the elements are in Z5.
27. Other matrix terminology:
All these terms are nicely explained on pp 49-50 of Hubbard. • transpose
• symmetric matrix • antisymmetric matrix • diagonal matrix
• upper or lower triangular matrix Try applying them to some 3 × 3 matrices: A = 3 1 2 1 2 3 2 3 4 B = 3 0 0 1 2 0 2 3 4 C = 3 1 2 0 2 3 0 0 4 D = 3 0 0 0 2 0 0 0 4 E = 0 −1 −2 1 0 −3 2 3 0
