- Syllabus (online)
- Syllabus (PDF)
- Final Examination, Wednesday, May 12, 8 AM
- Problem List [HTML][PDF]
- Course Projects (PDF)
- Grades: Advanced Linear Algebra (Math 390, Spring 2021) [Projects]

**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.

**SCLA**: A Second Course in Linear Algebra [PDF] (in-progress, updated 2021-01-16)**FCLA**: A First Course in Linear Algebra [390-Only Online Version]**TB**: Numerical Linear Algebra [Author's Page]**SW**: A Guide to Advanced Linear Algebra**GVL**: Matrix Computations (Fourth Edition)**HJ**: Matrix Analysis (Second Edition) [Errata]**DW**: Fundamentals of Matrix Computations (Third Edition)**HLA**: CRC Handbook of Linear Algebra (Second Edition)

**FCLA**: Section VS- Transcript

- Subspaces, span, linear independence, basis, dimension
**FCLA**: Chapter VS**FCLA**(Column Vectors only): Chapter V**GVL**: Section 2.1**SW**: Chapter 1- Transcript

- Definition, injective, surjective, invertible, vector space isomorphism
**FCLA**: Chapter LT**GVL**: Section 2.1**SW**: Chapter 2- Transcript

- Vector representation, matrix representation, change of basis
**FCLA**: Chapter LT**Worksheet**: SLA MR (upload MR.ipynb into CoCalc)- Transcript

- Eigenvalues, computing without determinants
**FCLA**: Section EE- Sage Demo EE: [Jupyter Notebook] [PDF]
- Transcript

- Eigenvalues, theorems and properties
**FCLA**: Section PEE- Transcript

- Invariant Subspaces
**FCLA**: Section IS- Sage Demo IS: [Jupyter Notebook] [PDF]
- Transcript

- Similarity, diagonalization, upper-triangular matrices
**FCLA**: Section SD- Transcript

- Continued...
**FCLA**: Section SD- Sage Demo SD: [Jupyter Notebook] [PDF]
- Transcript

- Characteristic polynomial, algebraic multiplicity
**FCLA**: Section CP- Transcript

- Matrix-vector product, matrix multiplication, similarity and representations
**FCLA**: Chapter M**FCLA**: Section SD**FCLA**: Section CB**TB**: Matrix-Vector Multiplication- Transcript

**SCLA**: 1.2 Direct Sums**SCLA**: 1.3 Orthogonal Complements**SW**: Section 1.4- Transcript

**SCLA**: 3.2 Nilpotent Linear Transformations**SW**: Sections 5.6, 5.7- Sage Demo CP: [Jupyter Notebook] [PDF]
- Transcript

**SCLA**: 3.2 Nilpotent Linear Transformations**SCLA**: 3.3 Jordan Canonical Form**SW**: Sections 5.6, 5.7- Start on Exercise 17 in the problem list
- In-Person, no transcript

**SCLA**: 3.3 Jordan Canonical Form**SW**: Sections 5.6, 5.7- Exercise 12 (FCLA Exercise SD.M61) should now be straightforward, and easier than Exercise 17.
- Transcript

**SCLA**: 3.3 Jordan Canonical Form- Sage Demo JCF: [Jupyter Notebook] [PDF]
**SW**: Sections 5.6, 5.7- Transcript

- Minimal polynomial, Rational Canonical Form, ZigZag Form
- Sage Demo MP: [Jupyter Notebook] [PDF]
- Transcript

**SCLA**: 2.1 LU (Triangular) Decomposition- TB: Lecture 20
- Sage Demo LU (Math 290 Section PDM): [Jupyter Notebook] [PDF]
- Transcript

**SCLA**: 2.2 QR (Gram-Schmidt) Decomposition- TB: Lecture 6 (Projectors)
- TB: Lecture 7 (QR Factorization)
- Transcript

**SCLA**: 2.2 QR (Gram-Schmidt) Decomposition**SCLA**: 1.5 Reflectors- TB: Lecture 6 (Projectors)
- TB: Lecture 7 (QR Factorization)
- Transcript

**SCLA**: 1.5 Reflectors**SCLA**: 2.2 QR (Gram-Schmidt) Decomposition- TB: Lecture 6 (Projectors)
- TB: Lecture 7 (QR Factorization)
- Transcript

**SCLA**: 1.7 Normal matrices**SCLA**: 1.8 Positive Semi-Definite Matrices**SCLA**: 2.3 Singular Value Decomposition- TB: Lecture 4 (The Singular Value Decomposition)
- TB: Lecture 5 (More on the SVD)
- TB: Lecture 31 (Computing the SVD)
- Transcript

**SCLA**: 1.8 Positive Semi-Definite Matrices**SCLA**: 2.3 Singular Value Decomposition- TB: Lecture 4 (The Singular Value Decomposition)
- TB: Lecture 5 (More on the SVD)
- TB: Lecture 31 (Computing the SVD)
- Transcript

**SCLA**: 2.3 Singular Value Decomposition- TB: Lecture 4 (The Singular Value Decomposition)
- TB: Lecture 5 (More on the SVD)
- TB: Lecture 31 (Computing the SVD)
- Transcript

**SCLA**: 2.3 Singular Value Decomposition- Sage Demo SVD Numerical Rank : [Jupyter Notebook] [PDF]
- Sage Demo SVD Image Compression (needs image) : [Jupyter Notebook] [PDF]
- Strang's 16 Factorizations (PDF)
- TB: Lecture 4 (The Singular Value Decomposition)
- TB: Lecture 5 (More on the SVD)
- TB: Lecture 31 (Computing the SVD)
- Transcript

**SCLA**: 2.3 Singular Value Decomposition- Sage Demo SVD Numerical Rank : [Jupyter Notebook] [PDF]
- Sage Demo SVD Visualization : [Jupyter Notebook] [PDF]
- TB: Lecture 4 (The Singular Value Decomposition)
- TB: Lecture 5 (More on the SVD)
- TB: Lecture 31 (Computing the SVD)
- Transcript

- Sage Demo Schur Decomposition : [Jupyter Notebook] [PDF]
**FCLA**: Section OD- Transcript

**FCLA**: Section OD- Transcript

- Sage Demo Cholesky Decomposition : [Jupyter Notebook] [PDF]
**SCLA**: Section 2.4 Cholesky Decomposition- Transcript

**SCLA**: (back to) Section 4.1 Least Squares- Transcript

- Michael Doob's: A deeper investigation of the properties of determinants
- Transcript

- Michael Doob's: A deeper investigation of the properties of determinants
- Transcript

- Michael Doob's: A deeper investigation of the properties of determinants
- Transcript

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

- CoCalc
- Sage Cell Server (one-off computations)

Main website for Sage: Sage Website

- MIT Beamer Tutorial [TeX Source!]
- Beamer User Guide [Documentation Directory (w/ theme examples)]
- Technically Speaking Videos
- Open Licenses Explained

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.

- 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

- 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

- 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