Advanced Linear Algebra (Math 420, Spring 2014)

Course Information

• Syllabus [PDF]
• Syllabus [HTML]
• Projects [PDF] [HTML]


• Problem List [HTML][PDF]


• Grades: Advanced Linear Algebra (Math 420, Spring 2014) [Projects]


• Course Research Guide @ Collins Library

Books

• SCLA: A Second Course in Linear Algebra [Provisional PDF]
• GVL: Matrix Computations (Fourth Edition)
• TB: Numerical Linear Algebra
• FCLA: A First Course in Linear Algebra [Online Version]
• HJ: Matrix Analysis (Second Edition) [Errata]
• SW: A Guide to Advanced Linear Algebra
• HLA: CRC Handbook of Linear Algebra (Second Edition)
• DW: Fundamentals of Matrix Computations (Third Edition)

Sage

• SageMathCloud
• UPS Sage Server (on-campus, VPN, proxy)

Presentation Advice

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

Course Calendar

• Course Introduction (Wednesday, January 22)

• MathSciNet (Friday, January 24)

  • With UPS Librarian, Eli Gandour-Rood

• Vector Spaces (Monday, January 27)

  • FCLA: Chapters V, VS
  • GVL: Section 2.1
  • SW: Chapter 1

• Linear Transformations (Wednesday, January 29)

  • FCLA: Chapter LT
  • GVL: Section 2.1
  • SW: Chapter 2

• Representations (Friday, January 31)

  • FCLA: Chapter R
  • GVL: Section 2.1
  • SW: Chapter 2
  • FCLA: See Solution MR.T80

• Direct Sums (Monday, February 3)

  • SCLA: Fundamentals/Direct Sums
  • GVL: Subsection 2.1.5
  • SW: Section 1.4

• Matrix Decompositions, LU Decomposition (Wednesday, February 5)

  • FCLA: Section FS, Extended Echelon Form
  • Handout: MAA Monthly, Four Subspaces and Extended Echelon Form
  • FCLA: Section OD, Triangular Matrices
  • SCLA: Matrix Decompositions/LU (Triangular) Decomposition
  • Worksheet: LU Factorization [SageMathCloud] [SageNB] [HTML] [PDF]
  • GVL: Chapter 3 (all of it)
  • GVL: For pivoting, Permutation Matrices, 1.2.8--12.11
  • TB: Lectures 20, 21, 22
  • DW: Sections 1.7-1.9 (note LDV decomposition)
  • IEEE 754
  • Goldberg, What Every Computer Scientist Should Know about Floating-Point Arithmetic

• LU Decomposition (Continued) (Friday, February 7)

• Numerical Linear Algebra (Monday, February 10)

  • TB: Appendix, The Definition of Numerical Analysis

• Householder Reflections (Wednesday, February 12)

  • SCLA: Fundamentals/Householder Reflections
  • GVL: Section 5.1
  • TB: Lecture 10
  • HJ: Section 2.1 (and facts about unitary matrices)
  • DW: Section 3.2

• Householder Reflections (continued) (Friday, February 14)

• QR Decomposition (Monday, February 17)

  • SCLA: Matrix Decompositions/QR Decomposition
  • GVL: Section 5.2
  • TB: Lectures 7, 8 and 10
  • HJ: Section 2.1
  • DW: Section 3.4

• QR Decomposition (continued) (Wednesday, February 19)

• Orthogonal Complements (Friday, February 21)

  • SCLA: Fundamentals/Orthogonal Complements
  • GVL: Subsection 2.1.5
  • TB: Lecture 2 (partially)
  • HJ: Section 0.6.6
  • DW: Section 3.5 (early)

• Application: Least-Squares Solutions (Monday, February 24)

  • SCLA: Applications/Least Squares
  • GVL: Subsection 5.3
  • TB: Lecture 11
  • HJ: Section 7.4.3 (with SVD)
  • DW: Section 3.1, 3.3

• Least Squares (continued) (Wednesday, February 26)

  • QR/Sage due before class. Sage Notebook to email announced in class.

• Q & A (Friday, February 28)

  • QR Sage Demo [Sage Worksheet] [PDF]
  • Discussion, Problem List

• Quiz 1 (Monday, March 3)

• Jordan Canonical Form (Wednesday, March 5)

  • SCLA: Invariant Subspaces

• Jordan Canonical Form (Continued) (Friday, March 7)

  • SCLA: Invariant Subspaces
  \

• Jordan Canonical Form (Continued) (Monday, March 10)

  • SCLA: Generalized Eigenspaces

• Jordan Canonical Form (Continued) (Wednesday, March 12)

  • SCLA: Generalized Eigenspaces

• Jordan Canonical Form (Continued) (Friday, March 14)

  • Project Proposals Due, 2 PM, on paper
  • SCLA: Generalized Eigenspace Decomposition

• Jordan Canonical Form (Continued) (Monday, March 24)

  • SCLA: Nilpotent Linear Transformations

• Jordan Canonical Form (Continued) (Wednesday, March 26)

  • SCLA: Canonical Form for Nilpotent Linear Transformations

• Jordan Canonical Form (Concluded) (Friday, March 28)

  • SCLA: Jordan Canonical Form
  • SW: Sections 5.6, 5.7
  • Worksheet: Jordan Canonical Form [SageMathCloud] [SageNB] [HTML] [PDF]

• Singular Value Decomposition (Monday, March 31)

  • SCLA: Fundamentals/Positive Semi-Definite Matrices
  • SCLA: Decompositions/Singular Value Decomposition
  • TB: Lectures 4, 5
  • GVL: Section 2.4

• Singular Value Decomposition (continued) (Wednesday, April 2)

  • Worksheet: Visualizing the SVD [Sage Notebook Worksheet]
  • Application: Numerical Rank [Sage Notebook Worksheet]
  • Application: Image Compression [Sage Notebook Worksheet]

• More Decompositions (Friday, April 4)

  • Cholesky: TB, Lecture 23
  • Schur: FCLA, Section OD

• "Lights Out!" (Monday, April 7)

  • Guest Lecture, Prof. Gary Gordon, Lafayette College
  • Project Drafts Due, 2 PM

• Alternative Definitions of Determinants (Wednesday, April 9)

• Quiz 2 (Friday, April 11)

• Projects Begin (Monday, April 14)

• Quiz 3 (Wednesday, May 14, 4 PM)

• Curve Fitting, Projectors (On the bubble)

Projects

• Monday, April 14, Troy Cornelius, Fast Matrix Multiplication [PDF] [Presentation]


• Wednesday, April 16, Imanuel Chen, Tournament Matrices [PDF] [Presentation]
• Wednesday, April 16, Peter Bergene, Applied Fuzzy Linear Algebra: Input Output Analysis [PDF] [Presentation]


• Friday, April 18, Davis Shurbert, Multilinear Algebra for the Undergraduate Algebra Student [PDF] [Presentation]
• Friday, April 18, Tyler Ueltschi, Third-Order Tensor Decompositions and their Application in Quantum Chemistry [HTML] [PDF] [Presentation]


• Monday, April 21, Aidan Meacham, Image Compression [PDF] [Presentation]
• Monday, April 21, Cameron Braithwaite, General Inner Product and Fourier Series [PDF] [Presentation]


• Wednesday, April 23, Ross MacAusland, The Moore-Penrose Inverse and Least Squares [PDF] [Presentation]
• Wednesday, April 23, Anna Dovzhik, Coding Theory: Linear-Error Correcting Codes [PDF] [Presentation]


• Friday, April 25, Hunter Wills, Linear Algebra for Computer Vision [PDF] [Presentation]
• Friday, April 25, Stephen Gower, Netflix Prize and SVD [PDF] [Presentation]


• Monday, April 28, Matt Reid, Pivoting for LU Factorization [PDF] [Presentation]
• Monday, April 28, Andrew Doss, Solving Toeplitz Systems of Equations and Matrix Conditioning [PDF] [Presentation]


• Wednesday, April 30, Brian Whetter, Modules Over Principal Ideal Domains [PDF] [Presentation]
• Wednesday, April 30, Spencer De Chenne, Minimum Polynomials of Linear Transformations [PDF] [Presentation]


• Friday, May 1, Glenna Toomey, Rational Canonical Form [PDF] [Presentation]
• Friday, May 1, Becky Hanscam, Calculating Kinetic Data by Linear Least-Square Data Fitting Methods for an Elementary Chemical Reaction [PDF] [Presentation]


• Monday, May 5, Garrett Buffington, Polar Decomposition of a Matrix [PDF] [Presentation]
• Monday, May 5, Jack Gilbert, Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition [PDF] [Presentation]

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.

Theory

• Spencer De Chenne: Minimal Polynomials of Linear Transformations
• Brian Whetter: Modules over Principal Ideal Domains
• Cameron Braithwaite: General Inner Products
• Garrett Buffington: Polar Decomposition of a Matrix
• Glenna Toomey: Rational Canonical Form
• Imanuel Chen: Tournament Matrices
• Davis Shurbert: Multilinear Algebra
• Zig-Zag Form (see me for citation)

Practice

• Ross MacAusland: The Pseudoinverse
• Andrew Doss: Solving Toeplitz Systems and the Importance of Conditioning
• Aidan Meacham: Linear Algebra and Digital Images
• Matt Reid: Pivoting for LU Factorization
• Troy Cornelius: Fast Matrix Multiplication
• Jack Gilbert: Markov Chains, Doubly Stochastic Matrices
• QR via Rotators
• Numerical Stability of a Specific Algorithm (two students possibly)

Applications

• Becky Hanscam: Calculating Kinetic Constants by Least Square Curve Fitting Methods
• Hunter Wills: Computer Graphics and Computer Vision
• Stephen Gower: Netflix Prize and Singular Value Decomposition
• Anna Dovzhik: Linear Error-Correcting Codes
• Peter Bergene: Leontief Input/Output Models (Economics)
• Tyler Ueltschi: Tensor Decompositions in Quantum Chemistry
• Google Page Rank and the SVD
• Least Squares and GPS (North American Datum)
• Signal Processing
• Linear Algebra for Computer Graphics

Sage

Sage is open-source software for advanced mathematics. There are many ways to use it.

• Sage Math Cloud (experimental)
• UPS Sage Notebook Server (campus or VPN only)
• Worldwide Notebook Server (popular, often slow)
• Sage Cell Server (one-off computations)

Main website for Sage: Sage Website

This is: http://buzzard.ups.edu/courses/2014spring/420s2014.html
Maintained by: Rob Beezer
Last updated: March 24, 2014