

Contents


Week 1: 
Algebraic preliminaries : Semigroups, monoids, groups , rings, structure of rings, ideals, fields. 
Week 2: 
Algebraic preliminaries : Vector spaces, modules and algebras.Linear independence,basis,subspaces and subspace operations, modular distributive rule,linear transformations and their representations. (HOMEWORK 1) 
Week 3: 
Vector spaces : Matrices, geometry induced by linear transformations, norms in vector spaces. Permutations, elementary operations and determinants, minors and cofactors, inverse transformations. 
Week 4: 
Vector spaces : Projections, insertion maps, factor spaces, invariant subspaces and induced maps, calculation of supremal and infimal invariant subspaces. (HOMEWORK 2) 
Week 5: 
Generalized eigenspaces and their calculations. Jordan form for real eigenvalues and its calculation. 
Week 6: 
Jordan form for complex eigenvalues and its calculation. (HOMEWORK 3) 
Week 7: 
Minimal polynomial, cyclic transformations and subspaces. Calculation of maximally cyclic subspaces and companion form. 
Week 8: 
Cyclic index, invariant factors and the Rational Canonical form and its relation to Jordan Form. Midterm Exam. 
Week 9: 
Linear functionals and dual spaces. Annihilators and the geometry of the dual spaces. Dual maps. 
Week 10: 
Inner product spaces,Bessel and Schwartz inequalities, representation of linear functionals,singular value decomposition, Hermitian operators, quadratic forms. (HOMEWORK 4) 
Week 11: 
Convex Analysis : Convex sets, convex and affine combinations, convex cones, dual cones. 
Week 12: 
Convex Analysis: Separation, Farkas Lemma. (HOMEWORK 5) 
Week 13: 
Convex analysis: Extreme points and directions. Linear Programming problem. Representation. Duality. 
Week 14: 
Factorization of matrices. LUfactorization. Cholesky factorization. Householder transformation and QRfactorization. (HOMEWORK 6) 
Week 15*: 
 
Week 16*: 
Final Exam. 
Textbooks and materials: 
Advanced Linear Algebra, S. Roman Finite Dimensional Vector Spaces, P. Halmos 
Recommended readings: 
An In troduction To Linear Algebra, L. Mirsky Matrix Analysis, R.A. Horn ve C.R.Johnson Matrix Computations, G.H. Golub ve C.F. Van Loan. 

* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

