(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.

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/