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Lecturer(s)
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Plešinger Martin, doc. Ing. Ph.D.
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Course content
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1) Basic terminology (linear function, linear equation, system of two equations); an arithmetic vector (summation, scalar multiple, linear combination); a matrix (matrix vector product, matrix matrix product); square, rectangular, and triangular matrix 2) Linear (in)dependenc (of vectors, columns/rows in a matrix, equations); system of linear algebraic equations (SLAR); equvalent transformations and their matrix representations; Gaussian elimination, rank of a matrix, solvability of SLAR (Frobenius theorem); singular, nonsingular, and inverse matrix 3) Gaussian elimination as a LU decomposition of the nonsinguar regular matrix; pivotation 4) Linear vector space; an algebraic vector; norm of a vector (the lenth); dot product (angles between vectors, the ortogonality); the range and the null-space of a matrix; matrix norms, the condition number as a measure of non-singularity 5) Orthogonal and unitary matrices; Givens rotation in R^2 and R^n; Householder reflexion in R^n 6) Application of ortogonal matrices to canelling entries of a given vectors; QR decomposition using orthogonal matrices; Gram-Schmidt ortogonalization 7) A recapitulation before the mid-term test (with some extensions: symmetric positive definite (SPD) matrices and Cholesky decomposition) 8) Determinant of a matrix; selected theorem about determinants; Cramer's rule 9) A geometric interpretation of norms, dot products, determinants, ortogonal matrices, vector product; analytic equation of a line, a plane, and a hyperplane in R^n and their parametric equations; the distance between a point and (hyper)plane; the distance between two lines, etc. 10) Eigenvalue problem, charakteristical polynomial, the companion matrix, spectrum; roots of polynomials of degree 1, 2, 3, and 4; roots of polynomials of higher degree (the basic theorem of algebra) 11) Similarity transformation; Schur theorem; Schur decomposition; normal matrices and their eigenvectors 12) Eigenvalues of unitary (ortogonal) matices, (skew-)Hermitian ((skew-)symetric) matices, and HPD (SPD) matices; diagonalizable matices; spectral decomposition; left and right eigenvectors 13) Defective matrices; Jordan form; functions of matrices 14) A recapitulation before the final test
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Preparation for exam
- 56 hours per semester
- Home preparation for classes
- 72 hours per semester
- Class attendance
- 84 hours per semester
- Preparation for credit
- 28 hours per semester
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Learning outcomes
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Basics of linear algebra and an itroduction into the practically applicable matrix calculus. The course contains three main topics: * systems of linear algebraic equations, Gaussian elimination and LU decomposition; * orthogonal matrices, QR decomposition and Gramo-Schmidt ortogonalization; * eigenvalues and eigenvectors. The course also briefly touches determinants and selected applications in analytic geometry.
Theory, algorithms and applications of linear algebra. Matrix algebra, geometric interpretations.
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Prerequisites
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Secondary school maths.
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Assessment methods and criteria
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Oral exam, Written exam
Written and oral exam.
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Recommended literature
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Bican, L. Lineární algebra a geometrie. Praha, 2000. ISBN 80-200-0843-8.
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Duintjer-Tebbens, E. J. a kol. Analýza metod pro maticové výpočty, základní metody. Matfyzpress. 2012.
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Strang, G. Introduction to Linear Algebra. Wellesley-Cambridge Press, 2003. ISBN 0-9614088-2-0.
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Watkins D.S. Fundamentals of Matrix Computations.. Jon Wiley & Sons, NY, USA, 1991. ISBN 0-471-61414-9.
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