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 number | 2 |
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. |