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Course, academic year 2023/2024
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Linear Algebra and Geometry II - NALG002
Title: Lineární algebra a geometrie II
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 8
Hours per week, examination: summer s.:4/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Jiří Tůma, DrSc.
Classification: Mathematics > Algebra
Interchangeability : NMAG102
Is incompatible with: NMAI045, NMAG112, NMAG102, NUMP004, NUMP003, NALG004, NALG003, NMAF032, NMUE025, NMAI044, NMAF012
Is pre-requisite for: NALG023
Is interchangeable with: NMUE025, NALG004, NALG086, NMAG102, NMAF012, NUMP004, NMAG112, NMAF032, NMAI044
Annotation -
Bilinear and quadratic forms, polar bases. Unitary spaces, orthogonality, affine and euclidian spaces. Projective space, classification of quadrics with a special emphasis to conic sections in affine and euclidian spaces. Jordan normal forms of matrices.
Last update: G_M (11.10.2001)
Literature - Czech

1. L. Bican, Lineární algebra a geometrie, Academia Praha 2000, ISBN 80-200-0843-8

2. J. Bečvář, Vektorové prostory I, II, III, SPN Praha 1978, 1981, 1982

3. J. Bečvář, Sbírka úloh z lineární algebry, SPN Praha 1975

4. L. Bican, Lineární algebra, SNTL Praha 1979

5. L. Bican, Lineární algebra v úlohách, SPN Praha 1979

Last update: Zakouřil Pavel, RNDr., Ph.D. (05.08.2002)
Syllabus -

1. Bilinear forms. Symmetrical and antisymmetrical bilinear forms, analytical expression, the defect and the rank, polar basis and its finding.

2. Quadratic forms. Polar bilinear form, vertex, defect, the law of inertia of quadratic forms, classification of forms.

3. Unitary space. Scalar product, orthogonality, orthogonal complement, orthogonal and orthonormal basis and their searching, orthogonal matrices, orthonormal polar basis of a quadratic form, unitary mappings, the metrics induced by the scalar product.

4. Affine space. The notion of an affine space, subspaces, the description of the subspaces by the equations, mutual position of subspaces, trasformation of coordinates, the transversal of oblique lines, affine mappings, partition ratio.

5. Euclidian space. Cartesian system of coordinates, the angle of vectors and directions, the distance of points, the distance of parallel subspaces, the distance of oblique lines.

6. Projective space. The notion of a projective space, homogeneous coordinates, projective expansion of an affine space, colinear mapping, double partition ratio, quadrics in projective spaces, projective, affine and metric classification of quadrics, dual space, the principle of duality.

7. Normal form of a matrix. Simple Jordan matrix, Jordan matrix, the existence and the unicity of the Jordan normal form, the methods of evaluation.

Literature:

L. Bican: Linární algebra a geometrie, Academia Praha 2000.

Last update: T_KA (22.05.2002)
 
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