Effective September 2024

The Joint Program Examination consists of two written examinations which cover topics from certain graduate courses in Linear Algebra and Analysis.

The names of these examinations are as follows:

• Linear and Numerical Linear Algebra
• Mathematical Analysis

A detailed list of topics for either examination is provided by the Joint Program Committee and will be posted on-line.

The examination for each part is 3 1/2 hours long. These examinations are held on different days of the same week in May and August of each year. Each department may specify the time limit for students in their department.

Topics for Mathematical 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

• 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.