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Course, academic year 2024/2025
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Linear algebra I - NMTM103
Title: Lineární algebra I
Guaranteed by: Department of Mathematics Education (32-KDM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: RNDr. Martina Škorpilová, Ph.D.
doc. RNDr. Jindřich Bečvář, CSc.
Teacher(s): doc. RNDr. Jindřich Bečvář, CSc.
RNDr. Martina Škorpilová, Ph.D.
Incompatibility : NMUM103
Interchangeability : NMUM103
Is incompatible with: NMUM103
Is interchangeable with: NMUM103
In complex pre-requisite: MC260P01C, MC260P01M, MC260P02C
Is complex co-requisite for: MC260P112, MC260P28
Annotation -
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)
Course completion requirements -

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. (30.08.2024)
Literature -

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. (30.08.2024)
Requirements to the exam -

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. (30.08.2024)
Syllabus -

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. (30.08.2024)
 
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