COURSE DESCRIPTION

Course code

MAP1140

Kind of studies

common to all departments;

Course title

Algebra and Analytic Geometry

First name, surname and title of the lecturer/supervisor

Komisja programowa Instytutu Matematyki i Informatyki

First name, surname and title of the team's members

Lecturers of the Institute of Mathematics and Computer Science

Course structure

Form of class Lecture Problems classes Laboratory Project Seminar Number of points
Number of hours/week 2 1 2+2
Course grade based on Examination Test

Prerequisites

High school mathematics

Course description

Algebraic expressions. Mathematical induction. Analytic geometry on plane and in space. Conics. Matrices. Determinants. Systems of linear equations. Complex numbers. Polynomials.

Lectures

Contents of particular hours Number of hours
1. ALGEBRAIC EXPRESSIONS. Transformation of algebraic expressions. Algebraic identities.2
2. MATHEMATICAL INDUCTION. Newton binomial formula. Application of mathematical induction to verification of inequalities and identities. 2
3. ANALYTIC GEOMETRY ON PLANE. Vectors on the plane. Operations on vectors. Scalar product. Orthogonality.2
4. Equations of the line (normal, directional, parametric). Parallel and perpendicular lines. Distance between a point and a line.2
5. CONIC SECTIONS. Circle, ellipse, hyperbola and parabola. Equations and properties.2
6. MATRICES. Operations on matrices (addition, multiplication and multiplication by scalars). Transposition. Identity, diagonal and symmetric matrices.2
7. DETERMINANTS. Definition – Laplace expansion. Cofactor of an element of matrix. Determinant of transposed matrix. 2
8. Elementary transformations of a determinant. Cauchy theorem. Nonsingular matrix, inverse matrix. Computation of inverse matrix by cofactors.2
9. SYSTEMS OF LINEAR EQUATIONS. Cramer theorem. Homogeneous and nonhomogeneous systems. 2
10. Solving of arbitrary systems of linear equations. Gauss elimination – transformation to upper triangular matrix. The case of nonsingular triangular matrix.1
11. ANALYTIC GEOMETRY IN SPACE. Coordinate system. Operations on vectors in R3. Length and scalar product of vectors. Angle between vectors. Cross product and triple product of vectors – computing area and volume.2
12. Plane. Equations of a plane. Normal vector of a plane. Angle between planes. Mutual location of planes. Line in space. Line as intersection of two planes. Equations of a line. Mutual location of two lines. Distance between a point and a plane or a line.2
13. COMPLEX NUMBERS. Operations on complex numbers in algebraic form. Conjugate numbers. Modulus.2
14. Polar form of complex number. Principal argument. De Moivre formula. n-th roots of a complex number2
15. POLYNOMIALS. Operations on polynomials. Root of polynomial. Bezout theorem. Fundamental theorem of algebra. Decomposition of a plynomial into factors. Decomposition of rational function into a sum of simple real fractions.3

Problems classes

Contents of particular hours Number of hours
1. Exercises illustrating the material presented during the lectures.15

Material for self preparation

Basic literature

1. J. Klukowski, I. Nabiałek, Algebra dla studentów, WNT, Warszawa 2005.
2. T. Huskowski, H. Korczowski, H. Matuszczyk, Algebra liniowa, Wydawnictwo Politechniki Wrocławskiej, Wrocław 1980.
3. T. Jurlewicz, Z. Skoczylas, Algebra liniowa 1. Przykłady i zadania, Oficyna Wydawnicza GiS, Wrocław 2006.
4. W. Stankiewicz, Zadania z matematyki dla wyższych uczelni technicznych, Cz. A, PWN, Warszawa 2003.
5. T. Trajdos, Matematyka, Cz. III, WNT, Warszawa 2005.

Additional literature

1. G. Banaszak, W. Gajda, Elementy algebry liniowej, część I, WNT, Warszawa 2002.
2. B. Gleichgewicht, Algebra, Oficyna Wydawnicza GiS, Wrocław 2004.
3. T. Jurlewicz, Z. Skoczylas, Algebra liniowa 1. Przykłady i zadania. Oficyna Wydawnicza GiS, wyd. 11, Wrocław 2006.
4. E. Kącki, D. Sadowska, L. Siewierski, Geometria analityczna w zadaniach, PWN, Warszawa 1993.
5. A. I. Kostrikin, Wstęp do algebry. Podstawy algebry, PWN, Warszawa 2004.
6. A. I. Kostrikin (red.), Zbiór zadań z algebry, PWN, Warszawa 2005.
7. F. Leja, Geometria analityczna, PWN, Warszawa 1972.
8. A. Mostowski, M. Stark, Elementy algebry wyższej, PWN, Warszawa 1963.

Conditions required for a student to pass the course

Positive result of the exam.