Algebra I E

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The course will cover all basic concepts of algebra necessary for every physicist. Emphasis will be placed on presentation of applications of algebra to computational problems of mathematical analysis and physics.

First part, extended level.

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

1. A. Białynicki-Birula "Algebra"

2. A. Mostowski, M. Stark "Algebra liniowa"

After having completed the course student should:

a) know the notion of field of complex number and do calculations with complex numbers

b) understand the notions of a vector space, linear independance, basis

c) understand the notions of a linear mapping and a matrix

d) solve systems of linear equations

e) compute determinanta, find inverese matrix

f) understand the notion of the dual space and the dual map

g) understand the notion of the multilinear map

Midterms and written exam -- computational part;

oral exam --theoretical part.