Syllabus

Abstract Algebra 

1. Fundamentals: sets, relations, and functions, Cartesian products and operations, partial orderings, equivalence relations and partitions, the Axiom of Choice and Zorn's Lemma, cardinal numbers.
2. Integers: mathematical induction and the least number principle, congruence, Division Algorithm, unique factorization, greatest common divisor and least common multiple, Euclidean Algorithm.
3. Groups (basic theory): semigroups and monoids, various characterizations of groups, subgroups, normal subgroups, homomorphism, isomorphism, quotient groups, direct products and sums, cosets and counting, Lagrange's Theorem, subgroup generation, the Isomorphism Theorems, the Correspondence Theorem.
4. Examples of groups: permutation groups, groups of symmetries, matrix groups, dihedral and quaternion groups.
5. Permutation groups: Cayley's Theorem, permutations as products of disjoint cycles and consequences for the structure of permutation groups, permutations as products of transpositions, alternating groups, simplicity of A_n for n > 4.
6. Abelian groups: structure of cyclic groups, free abelian groups and the structure of finitely generated abelian groups, the Fundamental Theorem of Abelian Groups.
7. Groups: structure theory, group actions on sets, stabilizers, the class equation, generalizations of Cayley's Theorem, Cauchy's Theorem, Sylow Theorems, classification of finite groups of small cardinality.
8. Rings (basic theory): subrings, ideals, homomorphism, isomorphism, quotient rings, direct products and sums, isomorphism and correspondence theorems, division rings and fields, examples of rings, rings of endomorphisms of an abelian group, rings of matrices.
9. Rings (advanced theory): properties of ideals, maximal and prime ideals, the Chinese Remainder Theorem, integral domains, relationship between maximal ideals and fields and between prime ideals and integral domains, factorization in commutative rings, irreducible and prime elements, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings.

Linear Algebra

1. Matrix arithmetic, reduced row ecehelon form of a matrix, general solution to a system of linear equations.
2. Determinants and their properties.
3. Vector spaces: subspaces, basis, coordinate vectors, change of basis.
4. Linear transfomations: matrix of a transformation, kernel, range, rank, Dimension Theorem, linear functionals, dual basis.
5. Inner products: Cauchy-Schwartz inequality, orthonormality, Gram-Schmidt, projection, Hermitian adjoint of a matrix and transformation.
6. Eigenvalues, eigenvectors, characteristic polynomial, minimal polynomial, Cayley-Hamilton Theorem, algebraic and geometric multiplicity, diagonalization.
7. Unitary matrices and transformations, normal matrices and transformations, unitary diagonalization of normal matrices, Spectral Theorem, Schur's Unitary Triangularization Theorem.
8. Canonical forms: Jordan canonical form, rational canonical form, invariant factors, elementary divisors, Primary Decomposition Theorem.
9. Hermitian matrices, Rayleigh-Ritz Theorem, variational characterization of eigenvalues (min-max) and applications, positive-definite matrices.

Examinations