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:
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.

Enrollment :
Attended : Analytic geometry

Examination :
Passed : Analytic geometry

Mandatory course - Regular study - Mathematics Education