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 become familiar with the structure of a finite-dimensional vector space, matrix computation, and methods for solving systems of linear equations

COURSE DESCRIPTION AND SYLLABUS:
Vector spaces. Introductory examples of vector spaces. The space of position vectors. Collinear and coplanar position vectors. Definition of vector spaces. Vector spaces Rn and Cn , the space of polynomials Pn, spaces of sequences, and the space of matrices Mmn . Linear span of a set. Generating set. Linear dependence and independence. Finite-dimensional vector space. Basis and dimension of a space. Representation of a vector in a basis. Canonical bases for the spaces Rn , Cn, Pn and Mmn . Application to determining interpolation polynomials.
Subspaces. Definition and basic properties of subspaces. Subspaces of R2 and R3 . Examples of matrix subspaces. Intersection and sum of subspaces. Dimension of the sum of subspaces. Examples. Direct complement. Examples. Direct complements of matrix subspaces. Linear manifolds. Quotient space.
Matrices. Matrix space. Special types of matrices. Matrix multiplication. Writing matrix products in column and row form. Inverse matrix. Determinant. Elementary matrix transformations. Reduction of a determinant to triangular form. Laplace expansion of a determinant. Determinant and matrix invertibility. Binet-Cauchy theorem. Matrix inversion using the adjugate. Matrix rank. Equivalent matrices. Similar matrices. Rank and invertibility of a matrix. Gauss-Jordan method for matrix inversion. LU factorization.
Systems of linear equations. Homogeneous and non-homogeneous systems of linear equations. Geometric interpretation of systems with two and three unknowns. Cramer systems. Matrix form of a system. Kronecker-Capelli theorem. Elementary transformations and equivalent systems. Structure of the solution set. Relation between the solution sets of homogeneous and non-homogeneous systems. Gauss and Gauss-Jordan elimination methods. Examples. Application of LU factorization in solving systems. Applications of systems of linear equations in natural and technical sciences.

1. Linearna algebra, Lj. Arambašić, Element, Zagreb, 2022.
2. Linear algebra done right, S. Axler, Springer-Verlag, New York, 1997.
3. Linearna algebra, D. Bakić, Školska knjiga, Zagreb, 2008.
4. Linear algebra and its applications, D. Lay, S. Lay, J. McDonald, Pearson, 2016.
5. Linear algebra, a modern introduction, D. Poole, Brooks/Cole, 2011.
6. Linear algebra and its applications, G. Strang, Saunders College Publ, 1986.

Mandatory course - Regular study - Mathematics