The Master's Comprehensive Examinations in Mathematics is based on the materials in the courses Math 600 and Math 603.

Math 600 Syllabus

Recommended text: Principles of Mathematical Analysis by W. Rudin

Details:

• Review of properties of R (the reals)
• Review of usual norm and inner product in R^n, countable and uncountable sets.
• Metric spaces, open/closed sets, interior, closure, limit point, limit of a sequence.
• Compact, connected, and complete metric spaces.
• Continuous functions on metric spaces, continuous image of a compact/connected set, extreme value theorem on a compact set, intermediate value theorem on a connected set.
• Uniform continuity.
• Fixed points, Banach contraction principle, mention of Brouwer fixed point theorem.
• Uniform convergence of sequences and series on  metric spaces, Weierstrass M-test, term-by-term integration and differentiation.
• Power series, radius and interval of convergence.
• Equicontinuity, Arzela-Ascoli theorem (proof optional) with illustrations. 
• The Weierstrass approximation theorem on [a,b], mention of generalization to rectangles in R^n.
• Riemann-Stieltjes integration (optional).
• Differentiation of functions on R^n, inverse and implicit function theorems (proofs optional) with illustrations.
  
• Optional topics.

Note: Instructors may cover optional topics in the course. However, problems from `optional' topics will/should not be given in the MS exams.

Math 430/603 Syllabus

Recommended text: Matrix Analysis and Applied Linear Algebra by Carl D. Meyer, SIAM, 2000.

Emphasis is on general theory of matrices rather than on theory of numerical linear algebra, which is covered in Math 630.

Details:

• Matrix algebra, matrix multiplication and inversion, linear equations, reduced row echelon forms, consistency, and rank
• Symmetric and Hermitian matrices, partitioned matrices
• Subspaces, linear independence, basis, and dimension
• Range and nullspace, Rank and Nullity Theorem
• Least squares
• Generalized inverses
• Linear transformations, change of basis and similarity
• Inner product spaces, orthogonality, and Gram-Schmidt
• Orthogonal and unitary matrices
• Complementrary subspaces, orthogonal decomposition
• Determinants, Schur complements and Schur determinantal formula
• Eigensystems
• Spectral theorems for normal and for symmetric matrices
• Quadratic forms and their extrema, positive definite matrices
• Jordan canonical form

Other topics faculty may wish to cover in Math 430, but which will not be included on the Comprehensive Exams include:

• Functions of matrices, matrix derivatives, applications to linear systems of ODEs
• Kronecker products and their eigenproperties
• Markov chains