Math 318
Advanced Linear Algebra: Tools and Applications
Math 318 is an advanced linear algebra course to follow Math 308. The emphasis of this course is on a matrix as an operator and as a way to organize data, in contrast to Math 308 which is all about solving linear equations. The focus is on structure, geometry and applications . This class does not cover algorithms, and numerical aspects of linear algebra.
The first half of the class is about eigenvalues and eigenvectors . Students see a range of spectral results for different types of matrices such as nonnegative, positive, Markov, symmetric, and positive semidefinite, along with their applications. Orthogonality and projections are central tools throughout the course. The next big emphasis of the class is on singular values and the singular value decomposition (SVD) of a general matrix and all the connections and applications of the SVD. Besides these core topics, students also see new types of vector spaces such as vector spaces of polynomials (applications to interpolation and solving a polynomial equation), vector spaces over finite fields (application to error correcting codes), and complex vector spaces (application to Fourier analysis). The course is designed to interweave theory and applications in almost all weeks.
Sample Syllabus
- Eigenvalues and Diagonalization
- Permutations and Determinant
- Difference Equations
- Nonnegative, Positive and Markov matrices
- Application: Gogle page rank
- Orthogonality
- Projections
- Application: Linear Regression
- Symmetric matrices and Quadratic Forms
- Positive Semidefinite Matrices
- Application: Graph Laplacians and connectivity
- Application: Distance Realization
- Application: Spectral Clustering
- Polynomial Vector Spaces
- Application: Polynomial Interpolation
- Application: Solving a polynomial equation
- Singular Value Decomposition
- Application: Low rank approximation of matrices, Image compression
- Application: Best fit planes to rows and columns, Principal Component Analysis
- Vector spaces over finite fields
- Application: Error Correcting Codes
- Complex Vector Spaces
- Application: Fourier analysis