Algebra I E 1100-1Ind02

The course will present basic concepts of linear algebra together with necessary bacground in abstract algebra. Covered material will serve as basis for development of more advanced linear algebra, analytic geometry and abstract algebra in the second semester.

Program:
1. Basics of linear algebra
(concept of a field, the field of complex numbers, polynomial with coeffitients in a field, divisibility and division of polynomials, Euclid's algorithm, Bezout's theorem, roots)
2. Vector spaces
(vector space, linear independence, basis, dimension, subspaces, sums, direct sums)
3. Linear maps
(linear maps, kernel, range, special classes of linear maps (monomorphisms, epimorphisms, isomorphisms, projections), matrix of a linear map, linear maps of kn, systems of linear equations, elementary operations on matrices, column/row reduction of a matrix, different descriptions of subspaces, matrix of an operator - change of basis)
4. Elements of duality theory
(dual space, dual basis, canonical isomorphism with secon dual, dual/adjoint operator)
5. Multilinear algebra and determinants
(mulitlinear maps, tensor products, permutations, determinants, determinant of an operator, the inverse matrix, invertible operators)

There will be written and oral exams. To take the oral exam the student needs to score at least 50% of points in the written part. In order to take the written part 50% of points from mid-term tests must be scored.

May 2008, Piotr Sołtan