Advanced Linear Algebra (Math 390, Spring 2021)
Course Information
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
Course Introduction, Vector Spaces (Tuesday, January 19)
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
Problem Session (Tuesday, January 26)
Eigenvalues I (Thursday, January 28)
Eigenvalues II (Friday, January 29)
Invariant Subspaces (Monday, February 1)
Problem Session (Tuesday, February 2)
Similarity and Diagonalization (Thursday, February 4)
Similarity and Diagonalization (Friday, February 5)
Characteristic Polynomial (Monday, February 8)
Problem Session (Tuesday, February 9)
Matrices (Thursday, February 11)
Direct Sums, Orthogonal Complements (Friday, February 12)
Review, Problems (Monday, February 15)
Exam 1 (Tuesday, February 16)
Jordan Canonical Form (Monday, February 22)
Jordan Canonical Form (Tuesday, February 23)
Jordan Canonical Form (Thursday, February 25)
Jordan Canonical Form (Friday, February 26)
Rational Canonical Form (Monday, March 1)
Matrix Decompositions, LU Decomposition (Thursday, March 4)
QR Decomposition (Friday, March 5)
QR Decomposition, Householder Reflectors (Monday, March 8)
Problem Session (Tuesday, March 9)
Householder Reflectors, QR Decomposition (Thursday, March 11)
Normal Matrices, Positive Semi-Definite Matrices, SVD (Friday, March 12)
Positive Semi-Definite Matrices, SVD (Monday, March 15)
Problem Session (Tuesday, March 16)
SVD (Thursday, March 18)
SVD (Friday, March 19)
SVD (Monday, March 22)
Problem Session (Tuesday, March 23)
Orthonormal Diagonalization (Schur Decomposition) (Thursday, March 25)
Orthonormal Diagonalization (Schur Decomposition) (Friday, March 26)
Cholesky Decomposition (Thursday, April 1)
Least Squares (Friday, April 2)
Problem Session (Monday, April 5)
Exam 2 (Tuesday, April 5)
Least Squares (Thursday, April 8)
Least Squares (Friday, April 9)
Projectors (Monday, April 12)
Problem Session (Tuesday, April 13)
Projectors (Thursday, April 15)
Projectors (Friday, April 16)
Determinants with Permutations (Monday, April 19)
Problem Session (Tuesday, April 20)
Determinants with Permutations (Thursday, April 22)
Determinants with Permutations (Friday, April 23)
Determinants via Axioms (Monday, April 26)
Problem Session (Tuesday, April 27)
Presentations (Monday, May 3)
Presentations (Monday, May 4)
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: Strassen’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
Presentation Advice
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