Linear Algebra

To my students: the class is over, but you can peruse the course evaluations. You can also try to locate yourself in this video from the last day of our class (Wednesday April 30th, 2014): what we do in this life echoes on YouTube.

To other students: you must be cramming for an upcoming exam. First cut your teeth on the homework and practice problems, and use the solutions only if necessary. Once you’re confident, go straight to the exams. My students had 50 minutes for each midterm, but you should really try to get each one done within 40 minutes (all at once, no snack breaks!). Textbooks and calculators were neither necessary nor allowed. All the best!

To instructors: perhaps you’re organising your own course, or a single lecture. The lecture notes are the hand-written, scanned notes I used for each class. If I were teaching the course again, I would almost certainly move Lecture 11 (graphs and adjacency matrices) later in the semester and Lecture 29 (Jordan forms) earlier. Good luck!

Hand-written and scanned; use them as an outline for reviewing the material.

Solutions to Homeworks 1–5 were written by the TA, Hua Qiang. Homework 8 was never collected: it is meant for final exam practice.

Most problems in Dr. Kazdan’s first midterm from Spring 2013 should also be accessible; his exam says kernel for null space and image for column space.

See also Dr. Jerry Kazdan’s exam material from Spring 2013, at the bottom of this page.

We met on Mondays, Wednesdays and Fridays from 11 AM to 12 PM in Room A4 of the DRL building. The semester had three units: the algebra, geometry and mechanics of matrix–vector operations; eigen-decomposition of square matrices; and several important advanced topics, including modern applications of linear algebra to ranking, image compression and so forth.

The prerequisite was an introductory linear algebra course (Math 240 at Penn). The best way to check whether you are ready for this material is to try as many old Math 240 final exam problems as possible.

The main textbook was Gilbert Strang’s Introduction to Linear Algebra (4th edition), though every homework problem was written out in full, so you never need a copy. As a reference I recommend David Poole’s example-filled Linear Algebra: A Modern Introduction. A free online textbook, taking a slightly more abstract approach, is Sergei Treil’s Linear Algebra Done Wrong; I strongly urge you not to judge it by its title!

Here’s a graph (no, not the kind with nodes and edges) of how the class did:

Regardless of how you feel about your grade, or what you will retain in the years (months? days??) to come, keep in mind that the material we covered, especially near the end, is quite hard-core and current. Perron–Frobenius theory underlies almost all modern search engine algorithms. SVDs are used regularly for image compression. Billions of dollars are being funnelled into research which hopes to produce even faster Fourier transforms. Entire books are dedicated to the manifold applications of linear optimisation in economics. You’ve seen some things this semester. And hopefully, they’ve seen you too.

I taught the course as a postdoc at Penn, with Hua Qiang as TA. The Penn Math Department gave me a good teaching award for it.