Robert B. Kusner

Director of the Center

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

Introduction to Linear Algebra (Math 235)

Common Guidelines & Topics Schedule:

All nine learning sections of Spring 2017 Math 235 will follow the Common Guidelines & Topics Schedule

Our Learning Sections:

TuTh at 11:30AM-12:45PM (235.3) & 1:00PM-2:15PM (235.7) in Goessmann 51

Office Hours:

Immediately after my afternoon classes (about 2:15PM) in Goessmann 51; or in LGRT 1435G (office) or 1535 (lab) with an advance appointment via email: profkusner@gmail.com

Prerequisites:

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

Remarks:

We're looking for some class note-takers, particularly from Section 7 (a few folks from Section 3 have kindly volunteered).
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 your generous fellow 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.


MyMathLab:

To register for Math 235 Linear Algebra Spring 2017 Kusner Sections 3 & 7:
1. Go to www.pearsonmylabandmastering.com
2. Under Register, select Student.
3. Confirm you have the information needed, then select OK! Register now.
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8. On the My Courses page, select the course name Math 235 Linear Algebra Spring 2017 Kusner Sections 3 & 7 to start your work.

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