SylabUZ

Wygeneruj PDF dla tej strony

Algebra liniowa 1 - opis przedmiotu

Informacje ogólne
Nazwa przedmiotu Algebra liniowa 1
Kod przedmiotu 11.1-WK-MATP-AL1-Ć-S14_pNadGenEIR82
Wydział Wydział Matematyki, Informatyki i Ekonometrii
Kierunek Mathematics
Profil ogólnoakademicki
Rodzaj studiów pierwszego stopnia z tyt. licencjata
Semestr rozpoczęcia semestr zimowy 2018/2019
Informacje o przedmiocie
Semestr 1
Liczba punktów ECTS do zdobycia 6
Typ przedmiotu obowiązkowy
Język nauczania polski
Sylabus opracował
  • dr hab. Krzysztof Przesławski, prof. UZ
Formy zajęć
Forma zajęć Liczba godzin w semestrze (stacjonarne) Liczba godzin w tygodniu (stacjonarne) Liczba godzin w semestrze (niestacjonarne) Liczba godzin w tygodniu (niestacjonarne) Forma zaliczenia
Ćwiczenia 45 3 - - Zaliczenie na ocenę
Wykład 45 3 - - Egzamin

Cel przedmiotu

To equip students with knowledge concerning basic algebraic structures such as fields, groups, vector spaces.

Wymagania wstępne

Secondary school mathematics.

Zakres tematyczny

Lecture
Fields
1. Number fields. (2h)
2. Operations. Axiomatic definition of a field. (2h)
3. The field of rational functions (1h)
4. The field of residue classes modlo p; Fermat’s little theorem (3h)
5. Isomorphisms of fields; automorphisms. The characteristic of a field. (2h)
6. Complex numbers: conjugation, modulus, polar form, geometric interpretation of addition and multiplication, De Moivre’s theorem, roots of complex numbers. (4h)
7. Fundamental theorem of algebra. Algebraic and transcendental numbers. (basic information) (1h)
8. Noncommutative fields: quaternions. (only briefly; students have to expand their knowledge by self study)(1h)
Permutations
1. Definition of a group; examples. (1h)
2. Parity of a permutation; alternating groups. (2h)
3. Decomposition of a permutation into disjoint cycles; decomposition into transpositions. (1h)

Vector spaces
1. Definition of a vector space; examples. (1h)
2. Linear independence; subspaces and spanning sets; basis; the Steinitz exchange lemma; dimension.(2h)
3. Linear transformations; spaces of linear homomorphisms; isomorphisms; linear transformations between coordinate spaces and their matrices; matrix multiplication and composition of linear transformations; algebras over a field: algebras of linear endomorphisms. (3h)
4. Rank of a matrix; the kernel and image of a linear transformation. (3h)
5. Matrix of a linear transformation with respect to arbitrary bases. (2h)
6. Dual space; dual basis; double dual and the canonical isomorphism between a space and its double dual; transpose of a linear transformation; transposed matrix. (2h)
Determinants
1. Determinant of a square matrix; multilinearity of determinant. (2h)
2. Determinant of a product of two matrices; determinant of a linear endomorphism. (2h)
3. Laplace expansion; inverse of a matrix. (1h)
4. General linear group, special linear group; group of upper triangular matrices. (1h)
Systems of linear equations
1. Existence of solutions (2h)
2. Fundamental system of solutions; dimension of the space of solutions. (2h)
3. Form of the solution to the system Ax=b, when A is an invertible matrix. (1h)
4. Gauss-Jordan elimination. (1h)

Class
Fields
1. Rational and irrational numbers; examples. Number fields; examples. (3h)
2. Two-argument operations and their properties. (1h)
3. Modular computations: tables of operations, inverse elements; binomial coefficients (exercises with the use of mathematical induction); applications of Fermat’s little theorem. (2h)
4. Complex numbers: finding products of numbers, and the inverse and the canonical form of a number. (2h)
5. Finding the argument and the modulus of a number. Roots. (2h)
6. Solving equations with complex coefficients. (2h)
7. Class test. (2h)
Permutations
1. Finding products of permutations. Inverses. Decompositions of permutations into cycles and transpositions. The sign of a permutation. (4h)

Vector spaces
1. Examples of vector spaces (2h)
2. Verification of linear independency; bases (2h)
3. Calculating values of linear mapping. Finding the kernel and the image of a linear mapping in some simple cases. (4h)
4. An algorithm for finding the rank of a matrix. (2h)
5. Class test. (2h)
Determinants
1. Applications of 2x2 determinants: the area of a parallelogram and a triangle. (2h)
2. 3X3 determinants: the volume of a parallelepiped. (2h)
3. Calculating certain determinants of large size. (4h)
Systems of linear equations
1. Finding the inverse of a matrix. (1h)
2. Checking the consistency of a linear system (2h)
3. Finding a fundamental system of solutions by Gauss-Jordan elimination. (2h)
4. Class test. (2h)

Metody kształcenia

Traditional lecturing, solving problems under the supervision of the instructor.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu Symbole efektów Metody weryfikacji Forma zajęć

Warunki zaliczenia

1. Preparation of the students and their active participation is assessed during each class by their instructor.
2. Class tests with problems of diverse difficulty helping to assess whether a student achieved minimal outcomes.
3. Written examination: It consists of around 18 problems. Each problem consists of 2 or 3 statements. To solve a problem, one has only to decide whether the statements are true or false. For some of them, however, explanations are demanded.
Final grade = 0.4 x class grade + 0,6 x exam grade. In order to be allowed to take the exam a student has to have a positive class grade. In order to pass the exam a student has to have a positive exam grade.

Literatura podstawowa

1. Strang, Gilbert, Linear Algebra and Its Applications, Cengage Learning, 2005.

Literatura uzupełniająca

1. G. Birkhoff, S. Mac Lane, A Survey of Modern Algebra, A.K. Peters, 1997.

Uwagi


Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 07-07-2018 08:19)