# 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. Junfang Li, in Campbell Hall 491 for scheduling information, advice, and information.

### Spring 2020 Schedule

- Real Analysis: Tuesday, May 5, 8:30 a.m. - 12:00 p.m.
- Linear Algebra: Thursday, May 7, 8:30 a.m. - 12:00 p.m.

### Fall 2020 Schedule

- Real Analysis: Tuesday, September 8, 8:30 a.m. - 12:00 p.m.
- Linear Algebra: Thursday, September 10, 8:30 a.m. - 12:00 p.m.

## Exam Topics

### Real Analysis

- Lebesgue measure on R
^{1}: outer measure, measurable sets and Lebesgue measure, non-measurable 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*spaces for positive measures with^{p}*p*=1,2,...,∞, definition, completeness.- Product measure, Lebesgue measure on R
^{k}, Fubini's theorem.

### Linear Algebra

- Vector spaces over a field: subspaces
- quotient spaces
- complementary subspaces
- bases as maximal linearly independent subsets
- finite dimensional vector spaces
- linear transformations
- null spaces
- ranges
- invariant subspaces
- vector space isomorphisms
- matrix of a linear transformations
- 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
- Jordon canonical forms
- real and complex normed and inner product spaces
- Cauchy-Schwarz and triangle inequalities
- orthogonal complements, orthonormal 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