## Advanced Linear Algebra (Math 390, Spring 2021)

#### Books

SCLA and FCLA by Beezer;
TB and SW are a close fit to our course's goals;
GVL, HJ and DW are encyclopedic reference works;
HLA is meant to be a comprehensive, encyclopedic resource.

#### Course Calendar

##### Vector Space Properties (Thursday, January 21)
• Subspaces, span, linear independence, basis, dimension
• FCLA: Chapter VS
• FCLA (Column Vectors only): Chapter V
• GVL: Section 2.1
• SW: Chapter 1
• Transcript
##### Linear Transformations (Friday, January 22)
• Definition, injective, surjective, invertible, vector space isomorphism
• FCLA: Chapter LT
• GVL: Section 2.1
• SW: Chapter 2
• Transcript
##### Representations (Monday, January 25)
• Vector representation, matrix representation, change of basis
• FCLA: Chapter LT
• Worksheet: SLA MR (upload MR.ipynb into CoCalc)
• Transcript

#### Projects

• Monday, May 3, Anna Van Boven, The Use of Matrix Decompositions to Initialize Artificial Neural Networks [PDF] [Presentation]

• Monday, May 3, Tristan Gaeta, Data Analysis Using Matrix Decomposition [PDF] [Presentation]

• Tuesday, May 4, Hayden Borg, The Discrete Fourier Transform: From Hilbert Spaces to the FFT [PDF] [Presentation]

• Tuesday, May 4, Jack Ruder, Alternatives to the Naive Algorithm for Matrix Multiplication: Strassenâ€™s, Triangular Matrices, and Inversion [PDF] [Presentation]

#### Sage

Sage is open-source software for advanced mathematics. There are several ways to use it, here are two of the easiest.

Main website for Sage: Sage Website

#### Project Topics

These are simply suggestions. Some I know well, some I know little about. Some are excellent choices, some will be harder than others. And you are not limited to just these topics. Many of these have been done by UPS students before (see Math 420 Spring 2014), so you need to be sure your approach is substantially different.

##### Theory
• Zig-Zag Form (see me for citation)
• Minimal Polynomials of Linear Transformations
• Modules over Principal Ideal Domains
• General Inner Products
• Polar Decomposition of a Matrix
• Rational Canonical Form
• Tournament Matrices
• Multilinear Algebra
##### Practice
• QR via Rotators
• Numerical Stability of a Specific Algorithm (two students possibly)
• The Pseudoinverse
• Solving Toeplitz Systems and the Importance of Conditioning
• Linear Algebra and Digital Images
• Pivoting for LU Factorization
• Fast Matrix Multiplication
• Markov Chains, Doubly Stochastic Matrices
##### Applications
• Google Page Rank and the SVD
• Least Squares and GPS (North American Datum)
• Signal Processing
• Linear Algebra for Computer Graphics
• Calculating Kinetic Constants by Least Square Curve Fitting Methods
• Computer Graphics and Computer Vision
• Netflix Prize and Singular Value Decomposition
• Linear Error-Correcting Codes
• Leontief Input/Output Models (Economics)
• Tensor Decompositions in Quantum Chemistry

This is: http://buzzard.ups.edu/courses/2021spring/390s2021.html
Maintained by: Rob Beezer
Last updated: January 20, 2021