UCLA Basic Examination
UCLA Basic Examination syllabus
The UCLA Basic Examination syllabus (January 2019) covers one-variable and multivariable analysis, metric space topology, and linear algebra. It lists preparation texts including Tao, Gamelin–Greene, Edwards, Apostol, Rosenlicht, Rudin, Petersen, Lang, Hoffman–Kunze, Serre, and references in numerical analysis.
Linear algebra
Finite-dimensional vector spaces over \(\RR\) and linear maps between them: subspaces, quotient spaces, dimension, bases, matrix representations. Positive definite inner products, orthonormal bases, extensions of orthonormal subsets to orthonormal bases. Eigenvalues and eigenvectors of automorphisms. Characteristic polynomials.
References: M. Artin, Algebra (Prentice Hall, 1991), chapters 1, 3, 4; K. Hoffman and R. Kunze, Linear Algebra (Prentice-Hall, 1971), chapters 1–6.
Abstract algebra
Definitions of groups, rings, fields, and modules over a ring. Homomorphisms of these objects. Subgroups, normal subgroups, quotient groups. Cyclic groups, and the structure theorem for finite abelian groups. Ideals, prime and maximal ideals, and their quotients, with the examples \(\ZZ\), \(k[X]\), and rings of algebraic integers. Field extensions, splitting fields of polynomials, normal extensions.
References: M. Artin, Algebra, chapters 2, 10, 11, 12, 13, 14; I. Herstein, Topics in Algebra (Blaisdell, 1964).
Point-set topology
Open and closed sets, continuous functions. Connectedness, compactness, the Hausdorff property, normality. Metric spaces, \(\RR^n\), the Heine–Borel theorem.
Reference: [Munk00].
Calculus
The differential of a smooth map between open subsets of Euclidean spaces. The matrix of partial derivatives. The inverse and implicit function theorems. Multivariable Riemann integration.
References: W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, 1964); A. Browder, Mathematical Analysis: An Introduction (Springer, 1996).
Complex analysis
Holomorphic functions, the Cauchy integral formula, power series representations of holomorphic functions, radius of convergence, meromorphic functions, residues.
Reference: L. Ahlfors, Complex Analysis (McGraw-Hill, 1973), chapters 1–5.
Real analysis
Advanced calculus at the level of the books of W. Rudin and A. Browder listed under calculus.
Additional topics
Pointwise and uniform convergence of functions, equicontinuity, \(\ell^2\), \(L^2(S^1)\), Hilbert spaces, orthonormal bases.