1. komponenta

Lecture type	Total
Lectures	60
Exercises	45

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

COURSE AIMS AND OBJECTIVES: To familiarize students with standard techniques of linear algebra (operations with matrices and vectors, solving systems of linear equations, applications of determinants) and with basics of the vector space structure (basis, dimension, subspaces). To recognize the vector space structure in examples covered by the Analytic geometry course.

COURSE DESCRIPTION AND SYLLABUS:
I. Vector spaces
I.1. Introduction and motivation for the concept of vector and vector space (a connection to the systems of linear equations up to 3 unknowns and analytic geometry).
I.2. Basic algebraic structures. Binary operation. Groupoid. Group and Abelian group. Basic properties of the group and examples. Symmetric group. Ring, basic properties and examples. Field, basic properties and examples.
I.3. Definition of a vector space. Basic properties and examples. Linear combination.
I.4. Linear span. Spanning set. Finitely generated vector space. Linearly independent set.
I.5. Basis of a vector space. Uniqueness of basis representation. Reduction of a finite spanning set to a basis. Cardinalities relation between linearly independent sets and spanning sets in finitely generated vector spaces. Equal cardinality of bases. Dimension of a vector space. Finite-dimensional vector space. Expanding a linearly independent set to a basis of a finite-dimensional vector space.
I.6. Subspace of a vector space. Subspace criterion (closedness under linear combinations). Intersection and sum of subspaces. Direct sum. Dimensions relation between sum and intersection of finite-dimensional subspaces. Complement to a subspace.

II. Matrices
II.1. Definition of a matrix, basic terms and notation. Some special types of matrices. Addition of matrices and multiplication of matrices by a scalar. Vector space Mm,n(F). Matrix multiplication. Algebra Mn(F).
II.2. Inverse matrix. General linear group GLn(F). Elementary operations on rows and columns. Equivalence of matrices. Elementary matrices. Rank of a matrix. Canonical form of a matrix.
II.3. Further properties of the matrix rank. Characterization of regularity of a quadratic matrix using rank. Determination of the inverse matrix by elementary operations. Orthogonal matrices.

III. Systems of linear equations
III.1. The concept of a system of linear equations, solution of the system and solvability of the system. Matrix representation of a system. Necessary and sufficient condition for solvability - The Kronecker-Capelli theorem. Criterion for the uniqueness of a solution.
III.2. Homogeneous system. Solution space of a homogeneous system. General form of the solution of a nonhomogeneous system. Gaussian method of solving a system. The structure of the set of solutions, the dimension of the solution space of the
associated homogeneous system. Linear manifold.

VI. Determinants
VI.1. Introduction to the concept of a determinant. Parity of a permutation. Definition of a determinant. Basic properties of determinants.
VI.2. Determinant properties related to elementary operations on rows and columns. Characterization of regular matrices using determinants.
VI.3. Binet-Cauchy theorem. Laplace expansion. Inverse matrix formula. Cramer's rule.

1. Linearna algebra, Z. Franušić, J. Šiftar, PMF, online izdanje, 2022.
2. Linearna algebra, Lj. Arambašić, Element, Zagreb, 2022.
3. Linearna algebra i primjene, D. Bakić, Školska knjiga, Zagreb, 2021.
4. Zbirka zadataka iz linearne algebre s rješenjima, N. Bakić, A. Milas, PMF - Matematički odsjek i Hrvatsko matematičko društvo, Zagreb, 1995.

Mandatory course - Regular study - Mathematics and Physics Education