MATH 325: Linear Algebra

Description

The course introduces systems of linear equations and techniques of finding their solutions based on row operations and Gaussian elimination. Fundamentals of matrix algebra including matrix product and matrix inverses and their properties are studied. Abstract vector spaces are introduced together with linear combinations and independence, subspaces, basis, and dimension. Main examples, namely, Rn, real matrix and polynomial spaces, as well as function spaces in one variable are emphasized. The course teaches linear transformations between (abstract) vector spaces, and range and kernel culminating in rank-nullity theorem. Representation of  linear transformations by matrices and change of basis are introduced with an eye towards diagonalization. General (real) inner products are used as a gateway to symmetric positive (semi-)definite matrices, leading to orthogonality, orthogonal bases, orthogonal projections, Gram-Schmidt orthogonalization, and their applications. Eigenvalues and eigenvectors are covered after a light introduction to determinants and with applications of diagonalization in mind. Spectral decomposition/factorization of symmetric matrices is treated as a lead to Singular Value Decomposition and its applications.

Prerequisite

Calculus I (Math 226 or equivalent); Math 301 Proof and Exploration recommended as a concurrent course

Content

1. Solution of system of linear equations using Gaussian elimination
2. Matrix algebra, special matrices (diagonal, triangular, symmetric), matrix inverse and transpose
3. Abstract vector spaces and main examples (Rn, matrix and polynomial spaces, function spaces)
4. Linear dependence/independence, subspaces, their bases and dimension (rank)
5. Linear transformations of vector spaces,matrix representations, change of basis
6. Range and kernel of linear transformations, finding their bases, rank-nullity theorem
7. Real inner product spaces and positive (semi-) definite matrices,inner products on function spaces
8. Cauchy-Schwarz and triangle inequalities
9. Orthogonality, orthogonal bases, orthogonal projections, Gram-Schmidt orthogonalization
10. Determinants (light)
11. Eigenvalues, eigenvectors, eigenspaces and their bases
12. Diagonalization and spectral factorization of symmetric matrices
13. Singular Value Decomposition (SVD) and applications

Student Learning Outcomes

Upon successful completion of the course, the students will be able to:

1. Find solutions of systems of linear equations using Gaussian elimination
2. Use bases and orthonormal bases to solve problems in linear algebra
3. Find the dimension of subspaces such as those associated with matrices
4. Find eigenvalues and eigenvectors and use them in applications
5. Compute SVD and use them in applications

Suggested textbooks

1. Linear Algebra, Fourth Edition, by Jim Hefferon  (https://joshua.smcvt.edu/linearalgebra/)
2. Applied Linear Algebra, Second Edition, by Peter Olver and Chehrzad Shakiban

Date: January 23, 2023