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Course, academic year 2023/2024
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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
Teaching methods: full-time
Guarantor: RNDr. Martina Škorpilová, Ph.D.
doc. RNDr. Jindřich Bečvář, CSc.
Incompatibility : NMUM103
Interchangeability : NMUM103
Is incompatible with: NMUM103
Is interchangeable with: NMUM103
In complex pre-requisite: MC260P01M, MZ370P19
Is complex co-requisite for: MC260P112, MC260P28
Annotation -
Last update: RNDr. Jakub Staněk, Ph.D. (14.06.2019)
An introductory course in linear algebra (introduction to basic algebraic structures, vector spaces, homomorphisms, homomorphisms and matrices, systems of linear equations).
Course completion requirements -
Last update: RNDr. Martina Škorpilová, Ph.D. (14.09.2023)

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:

More information is on the page

Literature -
Last update: RNDr. Martina Škorpilová, Ph.D. (05.10.2022)

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.

Requirements to the exam -
Last update: RNDr. Martina Škorpilová, Ph.D. (14.09.2023)

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).

Syllabus -
Last update: RNDr. Martina Škorpilová, Ph.D. (05.10.2022)

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.

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