Math 5500: Advanced Linear Algebra
Fall 2008
 

Instructor        Bryan Shader
Office              Ross Hall 204
Office Hours  M 1-2, W 11-12, or by appointment
email               bshader@uwyo.edu
Phone             766 6826
 

Course description
An introduction to the theory of vector spaces and linear transformations from an algebraic point of view with applications to matrix theory.

Topics include vector spaces, linear transformations and operators, structure theorems and inner product spaces.

Expected Learning Outcomes
Students are expected to achieve mastery of the topics listed below. By the completion of this course, students are expected to be able to do the following:

• Know and use relevant definitions.

• State and apply the major theorems presented in the course, and understand their significance.

• Be familiar with a wide-range of examples that illustrate the important
  concepts.

• Apply the tools and theorems to help solve pertinent mathematical problems both in and outside the course.

• Write correct and detailed mathematical proofs of results in Linear Algebra

Topics

1. Vector spaces (Bases and Dimension, Subspaces & Quotient Spaces, Blinear and quadratic forms)

2. Linear Transformations (Linear functionals & dual space, rank & nullity, matrix representations)

3. Linear Operations (Similarity, Invariant subspaces, minimal & characteristic polynomials, determinants & traces, Eigenvalues, Eigenvectos, Eigenspaces and generalized Eigenspaces, Jordan Canonical Form)

4. Inner Product Spaces (Orthogonality, orthogonal projections, inequalities, spectral theory for Hermitian, Normal, Positive definite & unitary operators)

Grading
There will be regular homework assignments (roughly weekly) worth 30% of your grade and a midterm and final exam each worth 35 of your grade.

Proofs should be (a) written in complete mathematical sentences, (b) logically correct, (c) primarily each student's own work, (d) the result of several iterations that refine the
solution and exposition.