MATH-UA 140 Linear Algebra

Term: Fall 2021
Instructor: Dr. Simon Becker
Level: Undergraduate

Topics

Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer’s rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.

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.