M346 Applied Linear Algebra

Prerequisite and degree relevance: The prereqiusite is M341 (or M311) or M340L, with a grade of C- or better, or consent of the instructor.

Text: Lorenzo Sadun, Applied Linear Algebra; the Decoupling Principle, second edition.

Responsible Parties: Charles Radin and Lorenzo Sadun, May, 2008.

We expect students to have a good feel for manipulating matrices, especially
row reduction, but also taking determinants. We also expect students to have seen abstract vector spaces and linear transformations, but some rustiness is expected, and those topics should be reviewed. It is not assumed that students have seen eigenvalues and eigenvectors; those should be done from scratch.

This is a course in serious mathematics, not a cookbook. As such, results in lecture, and in the book, should generally be proved rigorously. However, it's not an intro-to-proof class, and is aimed at an audience of engineers, economists and physicists (as well as mathematicians), so *writing* proofs should only play a minor role in the problem sets and exams.

Detailed Syllabus:
This number of days in this syllabus is based on a TTh class.

 Chapter 1. The Decoupling Principle (one day)

  •     Exploration: Beats

Chapter 2. Vector Spaces and Bases (two days)

  •     Vector Spaces
  •     Linear Independence, Basis, and Dimension
  •     Properties and Uses of a Basis
  •     Exploration: Polynomials
  •     Change of Basis
  •     Building New Vector Spaces from Old Ones
  •     Exploration: Projections

Chapter 3. Linear Transformations and Operators (three days)

  •     Definitions and Examples
  •     Exploration: Computer Graphics
  •     The Matrix of a Linear Transformation
  •     The Effect of a Change of Basis
  •     Infinite Dimensional Vector Spaces
  •     Kernels, Ranges, and Quotient Maps

Chapter 4. An Introduction to Eigenvalues (four days)

  •     Definitions and Examples
  •     Bases of Eigenvectors
  •     Eigenvalues and the Characteristic Polynomial
  •     The Need for Complex Eigenvalues
  •     Exploration: Circles and Ellipses
  •     When is an Operator Diagonalizable?
  •     Traces, Determinants, and Tricks of the Trade
  •     Simultaneous Diagonalization of Two Operators
  •     Exponentials of Complex Numbers and Matrices
  •     Power Vectors and Jordan Canonical Form

Chapter 5. Some Crucial Applications (five days)

  •     Discrete Time Evolution: x(n) = Ax(n   1)
  •     Exploration: Fibonacci Numbers and Tilings
  •     First Order Continuous Time Evolution
  •     Second order Continuous Time Evolution   
  •     Reducing Second Order Problems to First Order
  •     Exploration: Difference Equations
  •     Long Time Behavior and Stability
  •     Markov Chains and Probability Matrices
  •     Exploration: Random Walks
  •     Linear Analysis near Fixed Points of Nonlinear Problems
  •     Exploration: Nonlinear ODEs

Chapter 6. Inner Products (four days)

  •     Real Inner Products: Definitions and Examples
  •     Complex Inner Products
  •     Bras, Kets, and Duality
  •     Expansion in Orthonormal Bases: Finding Coefficients
  •     Projections and the Gram Schmidt Process
  •     Orthogonal Complements and Projections onto Subspaces
  •     Least Squares Solutions
  •     Exploration: Fourier Series
  •     Exploration: Curve Fitting
  •     The Spaces l2 and L2
  •     Fourier Series on an Interval

Chapter 7. Adjoints, Hermitian Operators, and Unitary Operators (three days)

  •     Adjoints and Transposes
  •     Hermitian Operators
  •     Quadratic Forms and Real Symmetric Matrices
  •     Rotations, Orthogonal Operators, and Unitary Operators
  •     Exploration: Normal Matrices
  •     How the Four Classes are Related Exploration: Representations Of SU2

Chapter 8. The Wave Equation (four days)

  •     Waves on the Line
  •     Waves on the Half Line: Dirichlet and Neumann Boundary Conditions
  •     The Vibrating String