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.