Robert B. Kusner

Director of the Center

(This page is like a cathedral: always under reconstruction ;-)

Kusner's Math Classes

About the photo: Rob's hair is now a lot greyer and a bit shorter, in part because he was elected to the Amherst Select Board several years ago (having also led a concerted and successful effort to preserve Amherst Town Meeting in 2005). However, unlike in mathematics, where truth is (effectively) eternal, in politics the voters decide: 2018 saw a slim majority adopt a new charter replacing Amherst's 259-year-old Town Meeting with a city form of government led by 13=(2+3i)(2-3i) Councilors. Rob was a candidate for one of the at-large Council seats: he made it through the 4 September 2018 primary, broke his leg in a bike accident on campus the next day, and 2 months later garnered 2114 votes (not enough to be elected, yet not bad for someone who could barely walk - let alone run). None of this should impact the mathematics you can learn with him if you come to class, pay attention, and ask good questions, but (occasionally) a wry comment about how well (or not) we are governing ourselves may slip from his lips....

And speaking of photos: here's a link to Rob's old class photos.

This Semester

Rob is on sabbatical for 2022-23....

Last Semester

Rob is on sabbatical for 2022-23....

How to Contact me

My office hours are always immediately after class, and by appointment on Zoom or in person – please email me at profkusner "at" gmail "dot" com to let me know you wish to discuss anything with me!

Last Year

In Spring 2022 it's often unclear how to prudently proceed (indoors? outdoors? remotely?) under continuing pandemic conditions and abruptly-changing guidelines, but with winter weather and the new omicron variants of COVID prevalent, we always do what makes sense for both public and personal health; whatever we do, please be sure to properly wear approved masks when indoors, as well as get vaccinated and boosted against COVID (and against the flu too)! Whenever we do need to "go remote" the Zoom invitation is shared (often a couple days, but at least a few hours) before the class, so please check your email early and often on class days!

Math 235

For students taking Rob's section Math 235-07 of the multisection course Introduction to Linear Algebra (which Rob is yet again, yet again chairing), the textbook remains Lay's Linear Algebra and Its Applications (5th edition) which you can inexpensively get on your own in .pdf or used-hardcopy format.

And for those still unenrolled: please be patient, attend class regularly, keep an eye on SPIRE for openings, and in a short time we should be able to officially enroll you (we don't want to deal with over-rides until we know they're necessary).

Here are instructions for how to get the Math 235 e-text and sign up for on-line homework/quizzes (Rob's section only). Assignments are due most Sundays at 11:59PM, but since the first isn't due till mid- February, if you're hesitating to "invest" in Pearson immediately, you can begin with the publisher's "free" offer: readable .pdf of the first two chapters of Lay's 5th edition.

Here is the course-wide Math 235 Topics Schedule and Common Guidelines page, which you should bookmark!

And here's a link to Math 235 Class Prep Videos on YouTube to watch before each class! You may also enjoy reviewing Basic Linear Algebra in 7 Easy Pages prepared by my former student (2011-14) Andrew Maurer (he recently earned his PhD at the University of Georgia; and his advisor Dan Nakano took my linear algebra course at Berkeley in the early 1980's - years later Dan told me that my course made him want to become a mathematician – perhaps the same is true for Andrew, but he's yet to confess ;-).

For you hardcore linear algebra students: the usual multiplication algorithm has complexity O(n^3) for n × n matrices; here's a semi-recent paper on the latest complexity bound: O(n^2.3726...). (Conjecture: O(n^2) is best.)

Note: The last part of the course uses the relation AE=ED (for any matrix E whose columns form a basis of real eigenvectors of a real 2 × 2 matrix A with distinct real eigenvalues) to factor A=EDE^{-1}, where D is the real diagonal 2 × 2 matrix whose diagonal entries are the eigenvalues, i.e. A is similar to D. This is useful to compute powers of A and the exponential of A. But what if A has repeated real eigenvalue (a, a), or if the eigenvalues are complex conjugate pairs (a+bi, a-bi) with nonzero b? How does one decompose A=aI+bJ where J^2=-I in the latter case, or decompose A=aI+N where N^2=0 in the former? Some texts (e.g. Bretscher) find a 2 × 2 matrix aI+bJ_o to which A is similar; here J_o is the standard 90-degree rotation matrix. There's another approach, suggested by my emeritus colleague Arunas Rudvalis, which seems simpler - and more general since it also deals with the N (nilpotent) case.

Welcome back from what we all hope was a wonderful winter break – please mask-up properly and get vaccine-boosted to stay healthy!

Important logistics for all classes

Accommodation Policy

UMass Amherst is committed to providing an equal educational opportunity for all students. A student with a documented physical, psychological, or learning disability on file with Disability Services may be eligible for academic accommodations to help them succeed in this course. If you have a documented disability that requires an accommodation, please notify me directly during the first two weeks of the semester so that we can make appropriate arrangements. Information on services and materials for registering with Disability Services are also available on the Disability Services website.

UMass Amherst's Official Academic Honesty Statement

Since the integrity of the academic enterprise of any institution of higher education requires honesty in scholarship and research, academic honesty is required of all students at the University of Massachusetts Amherst. Academic dishonesty is prohibited in all programs of the University. Academic dishonesty includes but is not limited to: cheating, fabrication, plagiarism, and facilitating dishonesty. Appropriate sanctions may be imposed on any student who has committed an act of academic dishonesty. Instructors should take reasonable steps to address academic misconduct. Any person who has reason to believe that a student has committed academic dishonesty should bring such information to the attention of the appropriate course instructor as soon as possible. Instances of academic dishonesty not related to a specific course should be brought to the attention of the appropriate department Head or Chair. Since students are expected to be familiar with this policy and the commonly accepted standards of academic integrity, ignorance of such standards is not normally sufficient evidence of lack of intent.



Rob taught and chaired Math 235: Introduction to Linear Algebra. This was our course-wide Math 235 F'21 Topics Schedule and Common Guidelines.

Rob also taught "Advanced" Linear Algebra (Math 545) in Fall 2021, asking his students to get a copy of the recommended textbook (Sheldon Axler's Linear Algebra Done Right) and begin to study it on theirr own (the best way to learn deeply) before the semester begins! (The required textbook is Gilbert Strang's Linear Algebra and Its Applications, 4th edition, and one can probably find cheap copies in abundance, but we used Axler's book at least as much.)

There were short weekly take-home quizzes and (roughly) bi-weekly problem sets following the 545f21topics schedule, a midterm exam (in-class, in late-October) and a final exam (3:30-5:30PM Wednesday 15 December 2021, also scheduled for our classroom); the course grade will be based on these three components (in roughly equal parts).


Rob continued remote-teaching from Coronavirus University (which – like Garrison Keillor's whimsical Lake Woebegone – is an institution where the all students are strong, all the administrators are good-looking, and all the (other) professors are above-average ;-)!

For those taking Linear Algebra (Math 235.06) with Rob, the incoming course chair more-or-less followed what Rob developed with the other instructors in Fall 2020.
While a lot of useful coursewide things are posted at our own learning section page (above), there is now also an official coursewide webpage for Spring 2021 Math 235.
Some of this information is mirrored at Moodle, where you'll also find Rob's Math 235-06 syllabus, but the pages above are the definitive ones.
For your convenience, here's where you'll find instructions for how to get the e-text and sign up for on-line homework (Rob's section only).

For students taking Honors Differential Geometry of Curves & Surfaces (Math 563H) with me, please have a look (below) at what we did last spring — the texts (required and recommended) remain the same.
I've begun posting the Math 563H ASCII Notes and HW Problems.
I'm also making more videos to share my own approach to curves and surfaces – I'll email students directly when I start to post these to YouTube, and later I'll curate a small library of my 563H videos (and others about differential geometry) – for now, just search the linked 563hw page for ""....
A bit of this information is also mirrored at Moodle....


Rob was chairing (again, yet again) and teaching Math 235: Introduction to Linear Algebra during the 2020 Fall semester. All students should have received email with instructions how to get the e-text and sign up for on-line homework for Rob's section (M235-03) only. Course-wide logistical details and important links appear here.


Rob taught Honors Differential Geometry of Curves & Surfaces (Math 563H). Here's a link to the S20 Math 563H ASCII Notes and HW Problems.

We moved to a remote format during spring break, sponsored by Rob's new institution Coronavirus University, courtesy of YouTube and Zoom. [So far, all the links were shared privately, but the best YouTube lectures and final projects will be posted here eventually!]

Less Recent

Rob chaired (yet again) and taught Math 235: Introduction to Linear Algebra during the Fall 2019 semester. Course-wide logistical details and links to individual learning sections appear here.

He also (again!) taught Advanced Multivariate Calculus (Math 425) in Fall 2019 [please see my comments below].

Advice (for those of you in Math 425):

The texts are as before: search for Calculus on Manifolds to find a .pdf of Spivak's beautiful text since it seems to be out of print (outside of China); the Marsden & Tromba text is a good resource and various recent editions should be available in used form; you should try to get - and begin reading! - both of these before the semester begins. And my former student, Kate Donoghue (now at grad school in Princeton) worked with me on a what we hope will become a helpful set of course notes and exercises - when ready, you'll find them here!

Before taking Math 425, students are expected to have a good understanding of basic linear algebra (Math 235) and vector calculus (Math 233); so if you're feeling a bit "rusty" in these areas, please "brush up" before the semester begins. Here's what I wrote to one prospective student:

Hi Dxxxx,

You should take a look at Marsden & Tromba's book and be sure you know about partial derivatives and multiple integrals; how to find critical points of functions of 2 or more variables, and test whether they're max, min, saddles or otherwise; and maybe a bit about finding critical points when the variables are constrained to the level set of another function (Lagrange multipliers). Although we'll (quickly!) review some of this in the first few weeks, and "remind" (I hope) you of the differential operators div, grad & curl, it's best if you've made an effort over the break!



Rob was chairing and teaching Math 235: Introduction to Linear Algebra during the 2019 Spring semester. This was how students got the e-text and sign up for on-line homework; course-wide logistical details appear here. Rob was also teaching Advanced Multivariate Calculus (Math 425) in Fall 2018.

Rob taught Advanced Multivariate Calculus (Math 425-01) in Spring 2018. He also was teaching a large section of Ordinary Differential Equations (Math 331-07), a course chaired by Jinguo Lian.

Rob also taught Advanced Multivariate Calculus (Math 425) in the Fall 2017 semester.

Rob chaired and taught Introduction to Linear Algebra (Math 235) during the 2017 Spring semester. Logistical details appear here.

Rob taught Advanced Multivariate Calculus (Math 425) in the Fall 2016 semester.

During AY 2015-16, Rob was on sabbatical: the winter, spring and summer at MSRI in Berkeley; and the fall at Penn, where he taught Geometric Variational Problems (Math 299/550), a topics course for advanced undergraduates and interested graduate students.

In the spring of 2015 Rob was teaching Honors Multivariable Calculus (Math 233H). Here's the ASCII page for Math 233H homework schedule and notes.

During the fall of 2014 Rob chaired - and taught two (double) sections (3 and 4) of - Introduction to Linear Algebra (Math 235). The next few paragraphs reflect past and foreshadow future versions of that course.

Here's a link to both our sections' S14 Math 235 WeBWorK. There will be roughly one problem set each week, usually due Tuesday evenings. Please start with the "Orientation" set to help you learn WeBWorK syntax. Over the years WeBWorK has improved, but occasionally you'll find a bug! If you're sure that you've figured out a problem, but WeBWorK won't swallow your answer, please don't waste time guessing and plugging -- each problem is "worth" so little anyway -- better if you've developed the skill and confidence to KNOW you've found a real bug, and not just made a silly mistake: such is life!

Here is the (draft ASCII) Math 235 (Linear Algebra) homework schedule for Rob's section.

Here's an evolving link to common course guidelines, suggested homework and practice exam problems.

During Spring 2014, Rob taught honors Differential Geometry of Curves & Surfaces (Math 563H).

Here is the (quick-loading, even on a slow connection) ASCII file for Math 563H (Curves and Surfaces) topics & problems. We are also working up a .tex/.pdf version....

Rob also taught (a double) section 2 of Introduction to Linear Algebra (Math 235) in Spring 2014.

In Fall 2013, Rob taught (a double) section 3 of Introduction to Linear Algebra (Math 235). Here's a link to Rob's F13 Math 235 WeBWorK.

Here's a link to Math 235 Common Midterm Review Problems which Prof. Paul Hacking will go over with you from 7-8:30 on Tuesday evening 12 November 2013 in Thompson 102.

Here are links to Math 235 Common Final Review Problems and Solutions which Prof. Paul Hacking went over from 7-8:30 on Tuesday evening 10 December 2013.

And here is a link to Fall 2013 Linear Algebra (Math 235) common guidelines.

In Spring 2013, Rob taught honors Differential Geometry of Curves & Surfaces (Math 563H). Here is the (quick-loading, even on a slow connection) ASCII file for Math 563H (Curves and Surfaces) topics & problems .

Rob also served as course chair for, and taught (a double) section 3 of Introduction to Linear Algebra (Math 235).

For the past several semesters, instructors have followed these Math 235 (Linear Algebra) common guidelines.

You can find some practice final exams below, but the following four may be the most helpful to review, along with your WeBWorK:

Math 235 Practice Final Problems from Fall 2008 , ... Spring 2009 , ... Fall 2009 and from Fall 2011 .

MAKE-UPS (only pre-approved according to UMass guidelines) will be handled by individual instructors.

Here's a link to the S13 235 midterm solutions (written up by Geri Jennings).

Here's a link to the S13 235 Mini-midterm2 solutions.

[Please continue to watch here for important developments and announcements!]

Rob was on leave at the Isaac Newton Institute for Mathematical Sciences for the Fall 2012 semester.

During Spring 2012, Rob is taught honors Differential Geometry of Curves & Surfaces (Math 563H). Here is the Spring 2012 ASCII file for Math 563H (Curves and Surfaces) topics & problems .

Rob also served as course chair for, and taught both (a double) section 4 and (a single) section 7 of Introduction to Linear Algebra (Math 235) in Spring 2012.

Here are notes on Inner Products, Gram-Schmidt and Ortho'matrices by Jenn Koonz (who gave a few lectures when Rob was in Germany at the end of March 2012) as well as notes on Determinants and on Eigenstuff by Arunas Rudvalis.

Here are links to Math 235 MiniMidtermII solutions (front) and (back) from Tuesday 17 April 2012.

And here are links to Math 235 Midterm solutions (page 1, 2, 3, 4, 5, and 6) from Wednesday 7 March 2012.

And links to Math 235 MiniMidtermI solutions (front) and (back) from Monday 13 February 2012.

In Fall 2011 Rob taught (a double) section 2 of Introduction to Linear Algebra (Math 235), with links to:

Fall 2011 Math 235 Practice Final Problems

Scanned pages of Fall Math 235 Common Midterm Solutions (find the sign error)!

Fall 2011 Math 235 Practice Midterm Problems.

Not as Ancient as You May Think...(Events of the Past Year Make Me "The Last Oak Standing Between Here and Eternity")

In Spring 2011, Rob taught the computational linear algebra (Math313/513-CS513) course while on sabbatical at Penn, and last fall he taught -- you guessed it! -- Introduction to Linear Algebra (Math 235) at UMass....

During Spring 2010, Rob taught honors Differential Geometry of Curves & Surfaces (Math 563H) and a double section of Linear Algebra (Math 235). Here is the ASCII file for Math 563H (Curves and Surfaces) topics & problems . (Andrew Reiter and Nicky Reyes are helping to TeX these now, so please stay tuned!)

Because he was the course chair during a semester when all the other instructors were graduate students, he and the other instructors established the following Math 235 (Linear Algebra) common guidelines, which may be useful to future course chairs.

Here's a link to the Fall 2009 Math 235 Practice Midterm and to sketches of answers! Here's a link to Fall 2009 Math 235 Midterm Answers. And here is a link to last year's Math 235 Practice Midterm. Finally, here's the Fall 2009 Math 235 practice final , along with Peter Norman's handwritten answers , a link to an old Math 235 practice final and links to older Math 235 practice finals or even older Math 235 practice finals, including one similar to our practice final with answers ....

Here are some Quick and Dirty Differential Equations notes prepared by Peter Norman last fall (with a little help from me) for use in Math 235.

During Spring 2009, Rob taught a special version of Math 462 (Geometry and the Imagination) - please visit here for the Math 462 topics and comments! He also taught a (double) section of Math 235 (Linear Algebra).

In Fall 2008, Rob taught a double-section of Math 235 (Linear Algebra).

Rob was on sabbatical in Spring 2008, but he taught back-to-back sections of Math 235 in Fall 2007 -- Linear Algebra is one of his favorite and most popular classes!

In Spring 2007 Rob taught Math 425 (Advanced Calculus) and a section of Math 235 (Linear Algebra). Here are links to the Math 425 (Advanced Calculus) homework schedule. Also, please have a look at my colleague Roman Federov's problems for enrichment and review. Rob organized the undergraduate TAP seminar (Math 191) during the Fall 2006 semester. It met in GANG (LGRT 1535) at lunchtime most Mondays and some Wednesdays. The following people kindly contributed to the 2006 TAP Topics. Many thanks to all of you!

In academic year 2006-07 Rob again planned to teach a two-semester course on Manifolds (Math 703-4) for graduate students. Due to unforeseen and unfortunate circumstances, Rob was not able to offer Math 704 in the spring. But here is where to find homework problems and informal notes for that course: Manifolds I (703) and Manifolds II (704), along with older notes from Fall 2002 (703), Spring 2003 (704) and Spring 2000 (Math 704).

In the Spring 2006 semester Rob taught Advanced Linear Algebra (Math 545) ; here are the updated problems/notes for this course. (Please compare with Rob's Spring 2002 advanced linear algebra problems/notes.) Rob also pinch-hit all semester for Ordinary Differential Equations (Math 331) in Spring 2006; here is the Math 331 general topics list and the Math 331 homework schedule for Rob's section. (This was an ambitious syllabus, covering more than the other sections did, and not without some ill-tempered jeers from the bleachers in the final innings -- it didn't help that Select Board meetings went on till after midnight on many, many occasions before this early morning course ;) In the Fall 2005 semester Rob taught Introduction to Abstract Algebra (Math 411); here are the notes and homework schedule .

In the Spring 2005 semester Rob was teaching an experimental version of Geometry and the Imagination. This was a very popular course, and Rob hopes to have the opportunity to offer it again soon, perhaps in conjunction with Math 563 (Differential Geometry of Curves and Surfaces)!

Rob was on sabbatical for 2003-04. During academic year 2002-03 Rob taught a two-semester course on Manifolds (Math 703-4) for graduate students. Here is where to find homework problems and notes from that course: Manifolds I (703) and Manifolds II (704), along with older notes from my Spring 2000 Math 704.

Rob has taught the basic graduate topology course several times, most recently in Fall 2000, Click here for the postscript or the rawtext version of the final Topology exam for Fall 2000.

And since this is Greek to me, here's a link to symbols like ε.


Watch here for further developments.


        1435G Lederle Graduate Research Tower
        Department of Mathematics
        University of Massachusetts at Amherst
        Amherst MA 01003

Phone 413 545 6022

Secretary 413 545 2762

Fax 413 545 1801

Electronic Mail profkusner "at" gmail "dot" com, kusner "at" math "dot" umass "dot" edu

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All material on this website is Copyleft* by Rob Kusner.

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