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.

Joint Program Examination Rules, effective September 2023 (PDF)

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:

  1. each exam may be attempted no more than twice, and
  2. 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 2022 Schedule

  • Real Analysis: Tuesday, September 6, 8:30 a.m. - 12:00 p.m.
  • Linear Algebra: Thursday, September 8, 8:30 a.m. - 12:00 p.m.

Exam Topics

  • Real Analysis
    • Lebesgue measure on R1: outer measure, measurable sets and Lebesgue measure, non-measurable sets, measurable functions.
    • The Lebesgue integral in R1: 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.
    • Lp spaces for positive measures with p=1,2,...,∞, definition, completeness.
    • Product measure, Lebesgue measure on Rk, Fubini's theorem.

    Topics for Analysis Joint Program Exam, Effective September 2023

    1. sup and inf for subsets of R, lim sup, lim inf for real sequences, Bolzano- Weierstrass theorem, Cauchy sequences.
    2. Continuous functions - min-max, intermediate value theorem, uniform conti-nuity, monotone functions.
    3. Derivative - mean value theorem, Taylor’s theorem for real functions on an interval.
    4. Riemann integration for functions on an interval. Improper integrals. Integrals depending on parameters.
    5. Sequences of functions - pointwise and uniform convergence, interchange of limits.
    6. Series of functions - M -test, differentiation/integration, real analytic functions.
    7. Metric spaces - open and closed sets, completeness and compactness, Cauchy sequences, continuous functions between metric spaces, uniform continuity, Heine-Borel and related theorems, contraction mapping theorem, Arzela-Ascoli 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
    • Cayley-Hamilton theorem and minimal polynomials, Jordan canonical form
    • real and complex normed and inner product spaces, Cauchy-Schwarz 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 Cn, 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.

Previous Exams

Previous Real Analysis and Linear Algebra exams will temporarily be housed in an exam testbank until they are transferred to the Canvas system.