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

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Linear Algebra 2 - opis przedmiotu

Cel przedmiotu

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.

Wymagania wstępne

Zapisz zmiany

Linear algebra 1.

Zakres tematyczny

Lectures:

Systems of linear equations:

1. Existence of solutions (2 hours)

2. Fundamental system of solutions; dimension of the solution space. (2 hours.)

3. The form of the solution for the system Ax=b when A is an invertible matrix. (1 hour)

4. Gaussian elimination method. (1 hour)

5. Characteristic equation; eigenvectors; eigenvalues; examples, applications (3 hours)

Jordan decomposition:

1. Algebraic sum of linear subspaces; direct sum. (1 hour)

2. Nilpotent linear endomorphisms; Jordan's block; invariant subspaces of an endomorphism (2 hours.)

3. Jordan decomposition of endomorphisms; Jordan form of the endomorphism matrix. (2 hours.)

Euclidean spaces:

1. The law of cosines - geometric definition of a dot product; dot product in Cartesian coordinates and its properties. (1 hour)

2. Formal definition of a dot product; norm of a vector; Schwarz inequality; angle between vectors, triangle inequality, parallelogram identity. (2 hours.)

3. Orthogonality: Pythagorean theorem, orthonormal basis (1 hour)

4. Gram-Schmidt orthogonalization procedure, existence of an orthonormal basis, vector in an orthonormal basis, orthogonal complement. (3 hours)

5. Isomorphism of Euclidean spaces; isomorphism of space and its dual (1 hour)

6. Adjoint linear operators; spectral theorem for self-adjoint endomorphisms. (3 hours)

Multilinear forms:

1. Multilinear mappings; multilinear forms: antisymmetric forms, symmetric forms.

2. Bilinear symmetrical forms; matrix of a bilinear form in a given coordinate system.

3. Diagonalization of the bilinear form; law of inertia for bilinear forms.

4. Quadratic forms; polarization formula - correspondence between bilinear and quadratic forms; normal form of a quadratic form. (5 hours)

 

Exercises:

Systems of linear equations:

1. Checking the consistency of a system of equations (2 hours)

2. Finding the fundamental system of solutions using the Gaussian elimination method (2 hours)

3. Eigenvalues - exercises (4 hours)

4. Test (2 hours)

Jordan decomposition:

1. Simple examples. Information about numeric packages. (2 hours.)

Euclidean spaces

1. Finding  the angle between vectors. Checking whether a given form is a dot product. (2 hours.)

2. Gram-Schmidt orthogonalization procedure, finding the orthonormal basis. Gram's determinant and its geometric interpretation. (5 hours)

3. Test. (2 hours.)

4. Diagonalization of simple self-adjoint mappings (4 hours)

Multi-line forms:

1. Bilinear form matrix; decomposition of a bilinear form into an antisymmetric and symmetric part. (1 hour)

2. Searching for the canonical (normal) form of a bilinear and of a quadratic form (2 hours)

3. Test (2 hours)

Metody kształcenia

Traditional lecture; auditorium exercises in which students solve exercises.

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 exercises. One exercise consists of several statements that are true must be resolved. 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 teacher may increase the grade by half a grade if he/she considers that the student deserves it (e.g. above-average student involvement; interesting solution to the exercises, 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

Zmodyfikowane przez dr Joanna Skowronek-Kaziów (ostatnia modyfikacja: 15-01-2024 17:42)