Math 196

MATH 19620 (Linear Algebra) is a linear algebra course taught at the University of Chicago, targeted primarily at economics and other social science majors. I taught the course twice, once in Spring 2012-13, and once in Autumn 2013-14 (these were successive academic quarters, with a summer break in between). I’ll include links to lecture notes and other material on this page. The course materials below are for the most recent version of the course I taught, namely the one in Autumn 2013-14.

Alternative learning recommendations

If you’re interested in recommendations for learning linear algebra, see Cognito Mentoring’s linear algebra learning recommendations page, to which I was a primary contributor.


The textbook for the course was Linear Algebra with Applications (5th Edition) (Featured Titles for Linear Algebra (Introductory)) [Hardcover] by Otto Bretscher. I’ll refer to it as “the textbook” or “the book” or “Bretscher” below and in the lecture notes and quizzes. However, the notes are largely independent of the textbook, and do not require you to buy or own the textbook. Having the 4th Edition will also suffice.

My personal teaching experience

You can read a detailed answer I wrote on Quora about my personal experience.

Full downloads

Later in this page, links to each of the lecture notes, quizzes, and review sheets are available. If, however, you want to download all the lecture notes or all the quizzes, saving each link can be a pain. The links below can be useful if you want to download in bulk. Note that page numbers as shown on the pages are for individual files, not for the combined file.

Opening remarks about linear algebra

Linear algebra: beware (PDF, 1 page)

Lecture notes plus quizzes

To determine the chronological order of quizzes, please use the dates in the quiz titles. Quizzes related to a given topic may not all have been administered at the time the topic was taught. Some quizzes were deliberately delayed in order to facilitate spaced repetition.

For solutions to any quiz, add -solns to the part of the URL just before the .pdf at the end of the URL. For instance, the solutions file for the quiz with URL has URL

Notes Sections in textbook (Bretscher) Related quiz(zes)
1.2, 1.3 Vectors: basic stuff
Linear functions: a primer Linear functions and equation solving (part 1), Linear functions and equation solving (part (2) (both shared with next lecture)
Equation solving with a special focus on the linear case Linear functions and equation solving (part 1), Linear functions and equation solving (part (2) (both shared with preceding lecture)
Gauss-Jordan elimination 1.2 Gauss-Jordan elimination, Matrix computations (shared with next)
Linear systems and matrix algebra 1.3 Linear systems, Matrix computations (shared with previous), Linear systems: rank and dimension considerations
Hypothesis testing, rank, and overdetermination (previous row)
Linear transformations 2.1 Linear transformations, (after basic matrix multiplication): Linear transformations and finite state automata
Matrix multiplication and inversion 2.3 and 2.4

Matrix multiplication: basic, as computational problems, abstract behavior prediction (do after finite state automata), rows, columns, orthogonality, and other miscellanea
Geometry of linear transformations 2.2 Geometry of linear transformations (abstract)
Image and kernel 3.1 Image and kernel: basic, computational, main
Linear dependence, bases, and subspaces 3.2, 3.3 Linear dependence, bases, and subspaces, Subspace, basis, and dimension
Coordinates (incl. similarity of linear transformations) 3.4 Similarity of linear transformations: main, applied
Abstract vector spaces 4.1 and 4.2 See “Applications to calculus” cluster below
Ordinary least squares regression 5.3 (unfaithful) OLS regression, Matrix transpose (preliminaries)

Quizzes providing glimpses into application areas not directly covered

Review sheets

Basic review sheets are unions of executive summaries of the corresponding lecture notes. Advanced review sheets include error-spotting exercises and other practice material.

Review sheet for … Sections of book covered
midterm 1 (basic) 1.1-1.3, 2.1
midterm 1 (advanced) 1.1-1.3, 2.1
midterm 2 (basic) 2.2-2.4, 3.1
midterm 2 (advanced) 2.2-2.4, 3.1
final (basic) 3.2-3.4, 4.1, 4.2, 5.3
final (advanced) 3.2-3.4, 4.1, 4.2, 5.3

Basic information