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### Linear algebra 1

 Code: 93076 ECTS: 9.0 Lecturers in charge: izv. prof. dr. sc. Zrinka Franušić Lecturers: izv. prof. dr. sc. Zrinka Franušić - Exercises Lukas Novak , mag. math. - Exercises English level: 1,0,0 All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.

### 1. komponenta

Lecture typeTotal
Lectures 60
Exercises 45
Description:
COURSE AIMS AND OBJECTIVES: The course covers the standard techniques in linear algebra and the basic notions of vector spaces (basis, dimension, subspaces).

COURSE DESCRIPTION AND SYLLABUS:
1. Definition of vector space over a field F. (Throughout the course F denotes R or C). Examples. Basic rules of computation. Linear combination. Linearly independent set. System of generators (spanning set).
2. Basis. Examples. Unique representation of a vector in a basis. Finitely generated vector spaces. Examples. Reduction of finite system of generators (spanning set) to a basis.
3. All bases are equipotent. Dimension. Examples. Extension of linearly independent set to a basis. Subspace. Examples: V2(O) in V3(O), space of solutions of homogeneous 3×3 system of equations in R3.
4. Intersection and sum of subspaces. Dimension of sum and intersection. Direct sum and direct complement. Examples: various complements for V2(O) in V3(O). Example: symmetric and anti-symmetric matrices in M3.
5. Vector space Mmn(F). Matrix multiplication. Algebra Mn(F). Regular matrices. Inverse matrix.
6. Determinant of a 2x2 system of linear equations.
7. Area of parallelogram equals det(u,v). General definition of determinant.
8. Elementary transformations of a determinant. Laplace expansion.
9. Adjugate matrix. Binet-Cauchy theorem. Characterization of regular matrices.
10. Rank of a matrix. Elementary transformations. Equivalent matrices.
11. Elementary matrices. Regularity and rank. Computing the inverse matrix by Gauss-Jordan transformations.
12. Systems of linear equations. Solvability and structure of the set of solutions. Linear manifold. Cramer's system. Gaussian elimination method.
13. Unitary spaces. Examples. Cauchy-Schwarz inequality. Norm. Orthonormal basis. Representation of a vector in an orthonormal basis. Gram - Schmidt orthogonalization.
14. Orthogonal complement. Least squares method. Approximate solutions of systems of linear equations.
Literature:
1. Z. Franušić, J. Šiftar: Linearna algebra 1, skripte dostupne na https://web.math.pmf.unizg.hr/~fran/predavanja-LA1.pdf
2. D. Bakić: Linearna algebra
3. N. Bakić, A. Milas: Zbirka zadataka iz linearne algebre
4. K. Horvatić: Linearna algebra
5. G. Strang: Linear algebra and its applications
Prerequisit for:
Enrollment :
Attended : Analytic geometry

Examination :
Passed : Analytic geometry
 2. semester Mandatory course - Regular study - Mathematics and Physics Education
Consultations schedule:

### Content

Link to the course web page: https://web.math.pmf.unizg.hr/nastava/ela/la1/