Qualifying Exams (Joint Program Examinations)
All Ph.D. students must pass two Joint Program Examinations. One exam covers Real Analysis (MA 645 and MA 646). The other exam covers Linear Algebra and Numerical Linear Algebra (MA 631 and MA 660). Each exam is three and a half hours long.
Master's degree students who are not planning on writing a thesis and who have passed the Joint Program Exam will not be required to take the final oral examination for the master's degree.
Scheduling an Exam
The examinations in Real Analysis and Linear Algebra are given during two periods each year (one in May and one in September). During each period a student may take one or both of the exams, subject to the following restrictions:
 each exam may be attempted no more than twice, and
 students may participate in exams during no more than three periods.
Any student considering taking an examination must meet as soon as possible with the Graduate Student Advisor, Dr. Roman Shterenberg, in University Hall 4035 for scheduling information, advice, and information.
Fall 2021 Schedule
 Real Analysis: Tuesday, September 7, 8:30 a.m.  12:00 p.m.
 Linear Algebra: Thursday, September 9, 8:30 a.m.  12:00 p.m.
Exam Topics

Real Analysis
 Lebesgue measure on R^{1}: outer measure, measurable sets and Lebesgue measure, nonmeasurable sets, measurable functions.
 The Lebesgue integral in R^{1}: positive functions and general functions, comparison with the proper and improper Riemann integral.
 Differentiation and integration: monotone functions, functions of bounded variation, absolute continuity, the fundamental theorem of calculus.
 Definition of a positive measure, measure spaces, measurable functions, the integral with respect to a positive measure.
 Convergence theorems for positive measures: monotone and dominated convergence.
 L^{p} spaces for positive measures with p=1,2,...,∞, definition, completeness.
 Product measure, Lebesgue measure on R^{k}, Fubini's theorem.

Linear Algebra
Linear Algebra
 Vector spaces over a field, subspaces, quotient spaces, complementary spaces
 bases as maximal linearly independent subsets, finite dimensional vector spaces
 linear transformations, null spaces, ranges, invariant subspaces, vector space isomorphisms
 matrix of a linear transformation, rank and nullity of linear transformations and matrices
 change of basis, equivalence and similarity of matrices, dual spaces and bases
 diagonalization of linear operators and matrices
 CayleyHamilton theorem and minimal polynomials, Jordan canonical form
 real and complex normed and inner product spaces, CauchySchwarz and triangle inequalities
 orthogonal complements, orthogonal sets, Fourier coefficients and the Bessel inequality
 adjoint of a linear operator, positive definite operators and matrices
 unitary diagonalization of normal operators and matrices, orthogonal diagonalization of real symmetric matrices
 bilinear and quadratic forms over a field
Numerical Linear Algebra
 Triangular matrices and systems, Gaussian elimination, triangular decomposition;
 the solution of linear systems, the effects of rounding error;
 norms and limits, matrix norms;
 inverses of perturbed matrices, the accuracy of solutions of linear systems;
 iterative refinement of approximate solutions of linear systems;
 orthogonality, the linear least squares problem, orthogonal triangularization, the iterative refinement of least squares solutions;
 the space C^{n}, Hermitian matrices, the singular value decomposition, condition;
 eigenvalues and eigenvectors, reduction of matrices by similarity transformations, the sensitivity of eigenvalues and eigenvectors;
 eduction to Hessenberg and tridiagonal forms;
 the power and inverse power methods, the explicitly shifted QR algorithm, the implicitly shifted QR algorithm;
 computing singular values and vectors, the generalized eigenvalue problem A−λB.