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2025/2026
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Welcome page > Faculty of Transportation Sciences > Department of Applied Mathematics > Linear Algebra

Linear Algebra

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Code Completion Credits Range Language
11LA Z,ZK 3 2P+1C+10B Czech
Course guarantor:
Martina Bečvářová
Lecturer:
Martina Bečvářová, Lucie Kárná, Pavel Provinský
Tutor:
Martina Bečvářová, Lucie Kárná, Pavel Provinský
Supervisor:
Department of Applied Mathematics
Synopsis:

Vector spaces (linear combinations, linear independence, dimension, basis, coordinates). Matrices and operations. Systems of linear equations and their solvability. Determinants and their applications. Scalar product. Similarity of matrices (eigenvalues and eigenvectors). Quadratic forms and their classification.

Requirements:

algebra and arithmetic (secondary schools level)

Syllabus of lectures:

1. Vector spaces and subspaces (linear combinations, linear independence, linear dependence, union of spaces, intersection of spaces, spanning set, properties of spanning set, dimension, basis, canonical basis, coordinates).

2. Matrices and operations (equal matrices, sum of matrices, matrix multiplication by scalars, matrix multiplication, commute matrices, elementary row operations, rank of matrix, diagonal matrix, transpose matrix, symmetric matrix, skew-symmetric ma-trix, triangular matrix, upper triangular matrix, lower triangular matrix, stairstep matrix, regular matrix, inverse matrix).

3. Systems of linear equations and their solvability, homogeneous systems of linear equations, non-homogeneous systems of linear equations, necessary and sufficient conditions for the existence of solution, structure of solutions, effective methods of solving. Matrix equations.

4. Determinants, methods of calculation, Laplace expansion, calculation of inverse ma-trix, Cramers rule. Determinants and their applications in algebra and geometry. Dot product, area and volume.

5. Similarity of matrices, eigenvalues, eigenvectors, eigenspace, generalized eigenvec-tors, Jordan block matrix, Jordan canonical form, transformations.

6. Quadratic forms, analytic expression, polar expression, polar basis, normal expression, canonical basis, classification of quad¬ratic forms, methods of classification, signature of quadratic forms, Sylvesters rule.

Syllabus of tutorials:

1. Vector spaces and subspaces (linear combinations, linear independence, linear dependence, union of spaces, intersection of spaces, spanning set, properties of spanning set, dimension, basis, canonical basis, coordinates).

2. Matrices and operations (equal matrices, sum of matrices, matrix multiplication by scalars, matrix multiplication, commute matrices, elementary row operations, rank of matrix, diagonal matrix, transpose matrix, symmetric matrix, skew-symmetric ma-trix, triangular matrix, upper triangular matrix, lower triangular matrix, stairstep matrix, regular matrix, inverse matrix).

3. Systems of linear equations and their solvability, homogeneous systems of linear equations, non-homogeneous systems of linear equations, necessary and sufficient conditions for the existence of solution, structure of solutions, effective methods of solving. Matrix equations.

4. Determinants, methods of calculation, Laplace expansion, calculation of inverse ma-trix, Cramers rule. Determinants and their applications in algebra and geometry. Dot product, area and volume.

5. Similarity of matrices, eigenvalues, eigenvectors, eigenspace, generalized eigenvec-tors, Jordan block matrix, Jordan canonical form, transformations.

6. Quadratic forms, analytic expression, polar expression, polar basis, normal expression, canonical basis, classification of quad¬ratic forms, methods of classification, signature of quadratic forms, Sylvesters rule.

Study Objective:

Learning basic concepts and methods of linear algebra and their applications in solving standard examples.

Study materials:

Blyth T.S., Robertson E.F., Matrices and Vectos Spaces, Essential Student Algebra, volume 2, Chapman and Hall, London, New York, 1986.

Blyth T.S., Robertson E.F., Linear algebra, Essential Student Algebra, volume 4, Chapman and Hall, London, New York, 1986.

Axler S., Linear Algebra Done Right, Undergraduate Texts in Mathematics, Springer, New York, Berlin, Heidelberg, 1996.

Curtis Ch.W., Linear Algebra. An Introductory Approach, Undergraduate Texts in Mathematics, Springer, New York, Berlin, Heidelberg, Tokyo, 1974 (2nd edition 1984).

Paley H., Weichsel P.M., Elements of Abstract and Linear Algebra, Holt, Rinehart and Winston, Inc., New York, Chicago, San Francisco, Atlanta, Dallas, Montreal, Toronto, London, Sydney, 1972.

Satake Ichiro, Linear Algebra, Pure and Applied Mathematics, A Series of Monographs and Textbooks, Marcel Dekker, Inc., New York, 1975.

Smith L., Linear Algebra, Undergraduate Texts in Mathematics, Springer, New York, Berlin, Heidelberg, 1978.

http://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra.html.

Note:
Further information:
https://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra.html
Time-table for winter semester 2025/2026:
Time-table is not available yet
Time-table for summer semester 2025/2026:
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The course is a part of the following study plans:
Data valid to 2025-05-26
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