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An introductory course in linear algebra (introduction to basic algebraic structures, vector spaces,
homomorphisms, homomorphisms and matrices, systems of linear equations).
Last update: Staněk Jakub, RNDr., Ph.D. (14.06.2019)
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Credit is a necessary and sufficient condition for taking the exam.
Credit exams practical knowledge and skills (numerical procedures, derivation, proving).
A prerequisite for obtaining credit is passing written test (one regular and two correction terms).
Another condition for granting the credit is participation in exercises (max. three absences; activity fulfilled by mastering specific tasks).
More information about credits is available at: http://www.karlin.mff.cuni.cz/~stepanov/
More information is on the page http://www.karlin.mff.cuni.cz/~becvar/
Last update: Škorpilová Martina, RNDr., Ph.D. (01.10.2025)
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R. A. Horn, Ch. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge, 2012.
S. Lang: Linear Algebra, Springer, New York, 2013.
I. Satake: Linear Algebra, Dekker, New York, 1975.
S. Axler: Linear Algebra Done Right, Springer, New York, 2015.
Last update: Škorpilová Martina, RNDr., Ph.D. (01.10.2025)
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The exam verifies theoretical knowledge (definitions, theorems), understanding mathematical derivation and proofs, formulation skills (using mathematical symbolism).
Credit is a necessary condition for taking the exam.
The structure of the exam (five questions): 1. definition and examples of defined term (2 points), 2. definitions and examples of defined term (3 points), 3. theorem (2 points), 4. simple proof of the given sentence (3 points) , 5. more difficult proof of the sentence (5 points).
The exam is written (approximaly 60 minutes), it is necessary to obtain at least 9 points (out of maximum 15 points).
The grade is determined by the points obtained for examination: 9-11 (Good), 12-13 (Very Good), 14-15 (Excellent). Last update: Škorpilová Martina, RNDr., Ph.D. (01.10.2025)
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Introduction to basic algebraic structures. Fields, rings, examples.
Vector spaces. Linear combinations, linear span, linear independence, generating sets, finitely and infinitely generated fields, basis, coordinates (with respect to a basis), dimension, theorem on the dimension of the join and meet; examples.
Homomorphisms of vector spaces. Basic properties of homomorphisms, special types of homomorphisms, the theorem on the dimension of the kernel and the image; examples.
Homomorphisms and matrices. The matrix of a homomorphism, compositions of homomorphisms and product of matrices, transformation of coordinates of a vector, rank of a matrix, elementary transformations, methods for calculating the rank of matrix, transformations of matrices, inverse matrix; examples.
Systems of linear equations. Solvability, the space of solutions and its dimension, the theorem of Frobenius, Gauss elimination method; problems; examples. Last update: Škorpilová Martina, RNDr., Ph.D. (01.10.2025)
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