(in Polish) Algebra liniowa 2400-ZL1AL
1. Systems of linear equations: elementary operations, equivalent systems, particular and general solutions, Gaussian elimination.
2. Vector spaces: examples, subspaces, linear combinations, linear independence, Steinitz lemma, bases and dimension, coordinates.
3. Linear transformations: examples; matrix representation in bases; operations on transformations and matrices; matrix algebra.
4. Determinants: properties, geometric meaning, computation.
5. Inverse and transpose; computing inverses.
6. Matrix rank and its relation to elementary operations and the determinant.
7. Using rank and determinant to solve systems; Kronecker–Rouché-Capelli theorem and Cramer’s rule.
8. Eigenvalues and eigenvectors; characteristic polynomial; eigenspaces; diagonal matrices and diagonalizability; criterion for diagonalizability.
9. Applications of diagonalization.
10. Standard inner product: norm, orthogonal/orthonormal systems and bases; Gram–Schmidt process; orthogonal projections and reflections.
11. Affine subspaces in Euclidean space; affine transformations; affine projections and reflections.
12. Quadratic forms and their matrices; Sylvester’s criterion; eigenvalue-based tests of (semi)definiteness.
13. Basics of linear programming; the simplex method.
Type of course
Course coordinators
Learning outcomes
Upon completion, the student:
1. Solves systems of linear equations (Gaussian elimination) and assesses uniqueness.
2. Tests linear independence and determines bases and dimension.
3. Computes matrix rank and determinant and interprets their meaning.
4. Finds eigenvalues/eigenvectors and diagonalizes when possible.
5. Uses the inner product for orthogonal projections and least squares.
6. Analyzes positive/negative (and semi-)definiteness of quadratic forms.
7. Formulates a basic linear programming model and solves it (e.g., simplex).
8. Uses computational tools (Octave/MATLAB) for the above tasks.
9. Interprets results in statistics, econometrics, and optimization, and verifies correctness.
Assessment criteria
Final grade = points from tutorial classes (quizzes/tests and participation; max 30) + points from a written final exam (max 70).
General
Bibliography
Koźniewski, Tadeusz. Wykłady z algebry liniowej I. Warszawa: Uniwersytet Warszawski, Wydział MIM, 2008.
Rutkowski, Jerzy. Algebra liniowa w zadaniach. 1. wyd. Warszawa: Wydawnictwo Naukowe PWN, 2012.
Antoniewicz, Ryszard; Misztal, Andrzej. Matematyka dla studentów ekonomii. Wykłady z ćwiczeniami. 4. wyd. Warszawa: Wydawnictwo Naukowe PWN, 2023.
Klukowski, Julian; Nabiałek, Ireneusz. Algebra dla studentów. 4. wyd. Warszawa: Wydawnictwo Naukowe PWN, 2021.
Banaszak, Grzegorz; Gajda, Wojciech. Elementy algebry liniowej. Cz. 1. Warszawa: Wydawnictwa Naukowo-Techniczne, 2002.
Banaszak, Grzegorz; Gajda, Wojciech. Elementy algebry liniowej. Cz. 2. Warszawa: Wydawnictwa Naukowo-Techniczne, 2002.
Lecture slides.
Octave documentation: https://docs.octave.org/latest/
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: