MATH 725: Advanced Linear Algebra

Prerequisites & Bulletin Description

Course Objectives

This is a second course in linear algebra in which students make the transition from Euclidean spaces and matrices to abstract vector spaces, inner product spaces and linear transformations. The emphasis is on axiomatic development, proof and conceptual understanding rather than calculation. Students will gain experience working abstractly without coordinates or determinants. In addition, they will learn how ideas from three dimensional geometry can be generalized to unify a wide variety of mathematical applications such as Fourier series, orthogonal functions and linear regression. This course should pave the way for further study in abstract algebra and advanced analysis. Upon successful completion, students will have a thorough understanding of the proofs of the finite dimensional versions of the Riesz Representation Theorem, the Spectral Theorem for normal operators, polar decomposition, singular value decomposition and the Jordan canonical form. They will also be able to apply the results to specific operators. 

Evaluation of Students

Instructors will design their own assessment schemes, which usually include weekly graded homework as well as midterm and final exams. 

Course Outline

Topics & Length
Topics Number of Weeks
Vector spaces, linear independence, bases, dimension 2
Linear maps, null space, range, matrix representations, invertibility.  2
Invariant subspaces, upper triangular matrix representations, eigenvectors and eigenvalues. 2.5
Inner products, norms, orthonormal bases, Gram Schmidt process, orthogonal projection and best approximation, linear functionals and adjoints, Riesz representation. 2.5
Self-adjoint and normal operators, the Spectral Theorem, positive operators, isometries, polar and singular-value decompostions. 3
Generalized eigenvalues, characteristic and minimal polynomials, nilpotent operators, Jordan canonical form. 3

Textbooks & Software

Linear Algebra Done Right, 2nd Edition, Sheldon Axler, Springer-Verlag, ISBN 0-387-98259-0

Finite Dimensional Vector Spaces, Paul Halmos, Princeton University Press (2001), ISBN 069-109-09-55

Linear Algebra 2nd Edition, Kenneth Hoffman, Prentice-Hall, ISBN 013-536-797-2

Submitted by: Eric Hayashi 
Date: June 3, 2003