SylabUZ
Nazwa przedmiotu | Algebra liniowa 2 |
Kod przedmiotu | 11.1-WK-MATP-AL2-Ć-S14_pNadGenINHLH |
Wydział | Wydział Matematyki, Informatyki i Ekonometrii |
Kierunek | Mathematics |
Profil | ogólnoakademicki |
Rodzaj studiów | pierwszego stopnia z tyt. licencjata |
Semestr rozpoczęcia | semestr zimowy 2018/2019 |
Semestr | 2 |
Liczba punktów ECTS do zdobycia | 6 |
Typ przedmiotu | obowiązkowy |
Język nauczania | polski |
Sylabus opracował |
|
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 | 30 | 2 | - | - | Zaliczenie na ocenę |
Wykład | 30 | 2 | - | - | Egzamin |
The objective of the whole course (linear algebra 1 and 2) is to prepare participants to self-study of theoretical and practical problems involving methods of linear algebra. The aim of each student should be to master the material included in the recommended book.
Linear algebra 1.
Lecture
Systems of linear equations
1. Characteristic equation; eigenvectors; eigenvalues; examples and applications. (4h)
Jordan decomposition
1. Algebraic sum of linear subspaces; direct sum. (1h)
2. Nilpotent endomorphisms; Jordan blocks. Invariant subspaces of an endomorphism. (2h)
3. Jordan decomposition of an endomorphism; Jordan normal form. (2h)
Euclidean spaces
1. Cosine theorem — geometric definition of a scalar product; scalar product in coordinate spaces. (1h)
2. Formal definition of a scalar product; norm; Schwarz inequality; angle between two vectors, triangle inequality; parallelogram law. (2h)
3. Orthogonality: Pythagorean theorem, orthonormal basis.(1h)
4. Gram–Schmidt algorithm, existence of an orthonormal basis, expansion of a vector with respect to an orthogonal basis, orthogonal complement. (3h)
5. Isomorphic Euclidean spaces; canonical isomorphism between a Euclidean space and its dual. (1h)
6. Conjugate of a linear transformation; spectral theorem for self-adjoint operations.
7. Orthogonal transformations; decomposition of a space into minimal invariant subspaces: rotations, reflections. Canonical matrix of an orthogonal transformation. Orientation.(5h)
Bilinear forms
1. Multilinear forms: skew forms, symmetric forms. (1h)
2. Bilinear symmetric forms: matrix of a form with respect to a given frame. (1h)
3. Diagonalization of a bilinear symmetric form; Sylvester’s law. (2h)
4. Quadratic forms; polarization formula — the one-to-one correspondence between symmetric and quadratic forms. (1h)
Class
Systems of linear equations
1. Solving eigenvalue problems. (4h)
Jordan decomposition
1. Simple examples. Information on numerical packages. (2h)
Euclidean spaces
1. Finding the angle between vectors. Checking whether a given form is a scalar product (2h)
2. Finding an orthonormal basis by Gram–Schmidt orthogonalisation process. Gram’s determinant and its geometrical interpretation. (5h)
3. Class test
4. Diagonalisation of simple self-adjoint transformations. (4h)
5. Classification of orthogonal transformations in dimensions 2 and 3. Composition of orthogonal transformations. Reduction of orthogonal matrices to their canonical forms – examples. (5h)
Bilinear forms
1. Matrix of a bilinear form. Decompsition of a form into skew and symmetric parts. (1h)
2. Diagonalization of bilinear forms (quadratic forms). (2h)
Traditional lecturing, solving problems under the supervision of the instructor.
Opis efektu | Symbole efektów | Metody weryfikacji | Forma zajęć |
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
1. Strang, Gilbert, Linear Algebra and Its Applications, Cengage Learning, 2005.
1. G. Birkhoff, S. Mac Lane, A Survey of Modern Algebra, A.K. Peters, 1997.
Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 07-07-2018 09:28)