1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Linear Algebra 1

Wygeneruj PDF dla tej strony

Linear Algebra 1 - opis przedmiotu

Cel przedmiotu

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

Wymagania wstępne

Zapisz zmiany

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.( 4 h)
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. (4h)
Determinants
1. Determinant of a square matrix; multilinearity of determinant. (3h)
2. Determinant of a product of two matrices; determinant of a linear endomorphism. (2h)
3. Laplace expansion; inverse of a matrix. (2h)
4. General linear group, special linear group; group of upper triangular matrices. (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, subspaces(4h)
2. Verification of linear independency; bases (4h)
3. Calculating values of linear mapping. Finding the kernel and the image of a linear mapping in some simple cases. (4h)
4. Class test. (2h)
5. An algorithm for finding the rank of a matrix. (2h)
6. An algorithm for finding the inverse matrix. (1h)

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)

 

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

The condition for passing the course is a positive grade in the exam. The exam consists of several tasks. One task consists of several statements whose truthfulness must be determined. Justification for selected statements should be provided: "This statement is true (false) because..." The form of the exam may change. The condition for taking the exam is a positive grade from the exercises. The grade for the course consists of the grade for the exercises (40%) and the grade for the exam (60%). The instructor may increase the grade by half a grade if he or she considers that the student deserves it (e.g. above-average student involvement; interesting solution to the task, etc.).

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

The description has been prepared originally by dr hab. Krzysztof Przesławski, prof. UZ.

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 21-02-2024 16:05)