MATH-UA 140 Linear Algebra

4 points. Fall and Spring terms.

Course Description

Linear algebra is an area of mathematics devoted to the study of structure-preserving operators on special sets (linear operators on vector spaces). Linear algebra is a cornerstone of any mathematics curriculum for two very important (and related) reasons:

  1. The theory of linear algebra is well understood and so a first step in many areas of applied mathematics is to reduce the problem into one in linear algebra.
  2. The spaces and operations studied in the subject are commonplace in many different areas of mathematics, science, and engineering.

Over the semester we will study many topics that form a central part of the language of modern science. The successful student will be able to:

  • Formulate, solve, apply, and interpret systems of linear equations in several variables;
  • Compute with and classify matrices;
  • Master the fundamental concepts of abstract vector spaces;
  • Decompose linear transformations and analyze their spectra (eigenvectors and eigenvalues);
  • Utilize length and orthogonality in each of the above contexts;
  • Apply orthogonal projection to optimization (least-squares) problems;
  • Explore other topics (as time permits).

The material we take up in this course has applications in physics, chemistry, biology, environmental science, astronomy, economics, statistics, and just about everything else. We want you to leave the course not only with computational ability, but with the ability to use these notions in their natural scientific contexts, and with an appreciation of their mathematical beauty and power.


  1. MATH-UA 121 Calculus I with grade of C or higher OR
  2. MATH-UA 211 Math for Economics I with grade of C or higher OR
  3. equivalent. Refer to the Calculus information page.

Sample Syllabi

Linear Algebra is not coordinated in the same sense as other multi-section courses with a common final exam (e.g., calculus). As such, the instructor has final discretion in topics chosen and course policies. Below are syllabi from two recent implementations.

Honors Linear Algebra

While the standard prerequisite for Linear Algebra I is ''Calculus I (MATH UA-121) or Math for Economics I (MATH-UA 211) or equivalent, with a grade of C or better, the prerequisites for this section are substantially greater. At minimum, students should have taken Calculus I or Math for Economics I or an equivalent class with a grade of A- or better. Students meeting this minimum may still find the honors section too difficult, since it assumes more mathematical maturity than is typically developed in just a year of AP calculus or a single semester of college-level calculus. Potential sources of the desired level of mathematical maturity include summer or after-school enrichment programs, or additional exposure to college-level mathematics. Prior exposure to linear algebra is not expected, though students with such exposure will naturally find it helpful.