# Preliminary Exam Syllabi

## Syllabus for the PhD Preliminary Examination in Analysis

### Topics

Metric spaces: open and closed sets, convergence, compactness, connectedness, completeness, continuity, uniform continuity, uniform convergence, equicontinuity and the Ascoli-Arzela Theorem, contraction mapping theorem

Single variable analysis: numerical sequences and series, differentiation, mean value theorem, Taylor's theorem, function series and power series, uniform convergence and differentiability, Weierstrass approximation theorem, Riemann integral, sets of measure zero

Several variables analysis: differentiability, partial derivatives, inverse and implicit function theorems, iterated integrals, Jacobians, change of variable in multiple integrals

Vector analysis: Stokes theorem, Green's theorem, divergence theorem

### References

1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, Freeman, 1993
2. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976
3. Michael Spivak, Calculus on Manifolds, Addison-Wesley, 1965
4. T. Apostol, Mathematical Analysis, Addison-Wesley, 1974

## Syllabus for the PhD Preliminary Examination in Linear Algebra

### Topics

Vector spaces: subspaces, linear independence, bases, dimension, isomorphism, linear functionals, dual space, adjoints, inverses and reducibility

Matrices and linear transformations: range, kernel, determinants, isomorphisms, change of basis, eigenvalues, eigenvectors, minimax Theory of eigenvalues, Gersgorin discs, minimal polynomial,
Cayley-Hamilton theorem, similarity, polar and singular value decomposition, spectral theorem, Jordan cannonical forms. Hermitian, symmetric, and positive definite matricies. Matrix andvector norms, convergence of sequences, powers etc. of matrices

Inner product spaces: inner products, norms, orthogonality, projections, orthogonal complement, orthonormal basis, Gram-Schmidt orthogonalization, linear functionals, isometries, normal operators, spectral theory, basic inequalities such as Cauchy Schwarz

### References

1. P.R. Halmos, Finite Dimensional Vector Spaces, Springer, 1993
2. R. Horn and C. Johnson, Matrix Analysis, Cambridge, 1999
3. Peter Lax, Linear Algebra, Wiley-Interscience, 1997.
4. K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice Hall, 1972
5. P. N. de Souza and J.-G. Silva, Berkeley Problems in Mathematics, Springer,
6. Berlin, 2004, (Chapter 7: Linear Algebra)