1. komponenta

Lecture type	Total
Lectures	60
Exercises	45

* Load is given in academic hour (1 academic hour = 45 minutes)

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:
I. VECTOR SPACES
I.1. Introduction and motivation for the concept of vector space (connection with systems of linear equations in 2 and 3 unknowns and analytic geometry). Binary operation. Groupoid. Basic algebraic structures. Group and Abelian group. Basic properties of a group. Examples. Symmetric group.
I.2. Ring, basic properties and examples. Field, basic properties and examples. Definition of vector space over a field F. Basic properties and examples. Linear combination.
I.3. Linear span. System of generators (spanning set). Finitely generated vector spaces. Linearly independent set. Basis. Unique representation of a vector in a basis. Reduction of finite system of generators (spanning set) to a basis. Relation between cardinalities of a linearly independent set and a system of generators. All bases of a finitely generated vector space are equipotent. Dimension of a vector space. Finite-dimensional vector space. Extension of a linearly independent set to a basis of a finite-dimensional vector space.
I.4. Subspace of a vector space. The subspace criterion. Intersection and sum of subspaces. Direct sum. Dimension of sum and intersection for finite-dimensional subpaces. Direct complement. Examples of decomposition as a direct sum of subspaces. Projection on a subspace along its direct complement.
II. MATRICES
II.1. Definition of matrix, basic notions and notation. Some special types of matrices. Addition of matrices and multiplication by a scalar. Vector space Mmn(F). Matrix multiplication. Algebra Mn(F).
II.2. Inverse matrix. General linear group GLn(F). Elementary transformations over columns and rows of a matrix. Equivalent matrices. Elementary matrices. Rank of a matrix. Canonical form of a matrix.
II.3. Further properties of rank of a matrix. Characterization of invertibility (regularity) by rank. Computing the inverse matrix by elementary transformations. Orthogonal matrices.
III. SYSTEMS OF LINEAR EQUATIONS
III.1.The notion of system of linear equations, solution and solvability of a system. Matrix form of a system of linear equations. Necessary and sufficient condition for solvability of a system (Kronecker-Capelli theorem). Condition for uniqueness of a solution of a system.
III.2. Homogeneous system of linear equations. Space of solutions of a homogeneous system. The form of the set of solutions of a general system of linear equations. Gaussian method for solving a system. Structure of the set of solutions. Dimension of the associated homogeneous system. Linear manifold.
IV. DETERMINANTS
IV.1. Introduction to the concept of determinant. Sign of a permutation. Definition of determinant of a square matrix. Basic properties of determinants. Further properties of permutations with regard to the sign. Properties of determinants related to elementary operations over rows and columns. Characterization of regularity of a matrix by its determinant.
IV.2. Binet-Cauchy theorem. Laplace expansion. Formula for the inverse matrix. Cramer's system.

1. Linearna algebra, D. Bakić, Školska knjiga, Zagreb, 2008.
2. Linearna algebra, K. Horvatić, Golden marketing - Tehnička knjiga, Zagreb, 2003.
3. Zbirka zadataka iz linearne algebre, N. Bakić, A. Milas, PMF - Matematički odjel, Zagreb, 1996.
4. Linearna algebra 1, skripte, Z. Franušić, J. Šiftar, PMF-MO.
5. Linear algebra and its applications, G. Strang, Saunders College Publ, 1986.

Enrollment :
Attended : Analytic geometry

Examination :
Passed : Analytic geometry

Mandatory course - Regular study - Mathematics Education