In this course we will explore aspects of linear algebra that are of particular use in concrete applications. For example, we will learn how to factor matrices in various ways that aid in solving linear systems. We will also learn how one can effectively compute estimates of eigenvalues when solving for precise ones is impractical. In addition, we will investigate the theory of quadratic forms. The course will involve a mixture of theory and computation. It is important to understand why our methods work (the theory) in addition to being able to apply the methods themselves (the computation). Students will be expected to write proofs on homework assignments and tests.
Official course description
Vector and matrix norms. Schur canonical form, QR, LU, Cholesky and singular value decomposition, generalized inverses, Jordan form, Cayley-Hamilton theorem, matrix analysis and exponentials of matrices, eigenvalue estimation and the Gershgorin Circle Theorem; quadratic forms, Rayleigh and minima principles. This course includes proofs and applications of computational methods.
Click the above header to download lecture notes for the course. They are a single file that may be updated as the course progresses. They should be complemented by your own class notes. I will often say more in class than is in the lecture notes.
The following textbooks are recommended as references for the course material. They can all be downloaded free of charge. They are listed in approximate order of usefulness for this course.
- Linear Algebra with Applications, W. Keith Nicholson
- A First Course in Linear Algebra, Kenneth Kuttler and Lyryx
- Linear Algebra Done Wrong, Sergei Treil
- Linear Algebra, David Cherney, Tom Denton, Rohit Thomas, and Andrew Waldron
- Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares, Stephen Boyd and Lieven Vandenberghe
There are two main locations for information regarding the course: