Robert B. Kusner

Director of the Center

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

Math 235: Introduction to Linear Algebra

• • • Here's how to get the e-text and sign up for on-line homework for Rob's sections (6 and 7) only!

• • • Course-wide logistical details appear here.

Common Guidelines & Topics Schedule:

All nine learning sections of Spring 2019 Math 235 will follow the Common Guidelines & Topics Schedule

Our Learning Sections:

TuTh at 1:00PM-2:15PM (235.07) & 2:30PM-3:45PM (235.06) in LGRT 121

If you are Waiting to Enroll:

Please come to class, introduce yourself, 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).

Office Hours:

Between (about 2:15PM) and immediately after (about 3:45PM) my classes in LGRT 121; and in LGRT 1435G (office) or LGRT 1535 (lab) with an advance appointment via email:


Familiarity with basic algebra, vector geometry and (later in the course) a bit of calculus and differential equations


We're always looking for some class note-takers! Please talk with Rob after class if you're interested in this important project.

Here's some possibly useful (or useless ;-) information:

⇒ Before the first midterm, you may enjoy reviewing Basic Linear Algebra in 7 Easy Pages prepared by my former student (2011-14) Andrew Maurer (he's now in grad school 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).

⇒ Before the second midterm, you may appreciate this Linear Algebra Review Sheet by a recent (Spring 2017) student Jonah Palmer (with a couple comments from yours truly).

⇒ Several past exams are available here or at the Kusner's Math Classes page.

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

⇒ 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 nonzero 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. Here'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.


To register for Math 235: Introduction to Linear Algebra - Spring 2019 (Kusner Sections 6 and 7 only!) here are instructions in .pdf or in text:
1. Go to
2. Under Register, select Student.
3. Confirm you have the information needed, then select OK! Register now.
4. Enter our learning sections' course ID: kusner50668, and Continue.
5. Enter your existing Pearson account username and password to Sign In.
→You have an account if you have ever used a Pearson MyLab & Mastering product, such as MyMathLab, MyITLab, MySpanishLab, MasteringBiology or MasteringPhysics.
⇒ If you don’t have an account, select Create and complete the required fields.
6. Select an access option.
⇒ Enter the access code that came with your textbook or was purchased separately from the bookstore.
⇒ Buy access using a credit card or PayPal account.
⇒ If available, get temporary access by selecting the link near the bottom of the page.
7. From the You're Done! page, select Go To My Courses.
8. On the My Courses page, select the course name Math 235: Introduction to Linear Algebra - Spring 2019 to start your work.

To sign in later:
1. Go to
2. Select Sign In.
3. Enter your Pearson account username and password, and Sign In.
4. Select the course name Math 235: Introduction to Linear Algebra - Spring 2019 to start your work.

To upgrade from temporary to full access:
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3. Enter your Pearson account username and password, and Sign In.
4. Select Upgrade access for Math 235: Introduction to Linear Algebra - Spring 2019.
5. Enter an access code or buy access with a credit card or PayPal account.

Accommodation Policy Statement:

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 your instructor during the first two weeks of the semester so that we can make appropriate arrangements. It is also your responsibility to notify Disability Services (normally at least one week in advance) if you wish to take an exam there.

!!!DRAFT!!! (Still under reconstruction! ;-)