Linear Algebra
Code  Completion  Credits  Range  Language 

11LA  Z,ZK  3  2P+1C+10B  Czech 
 Garant předmětu:
 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, skewsymmetric matrix, 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, nonhomogeneous 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 matrix, Cramer’s rule. Determinants and their applications in algebra and geometry. Dot product, area and volume.
5. Similarity of matrices, eigenvalues, eigenvectors, eigenspace, generalized eigenvectors, 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, Sylvester’s 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, skewsymmetric matrix, 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, nonhomogeneous 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 matrix, Cramer’s rule. Determinants and their applications in algebra and geometry. Dot product, area and volume.
5. Similarity of matrices, eigenvalues, eigenvectors, eigenspace, generalized eigenvectors, 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, Sylvester’s 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
 Timetable for winter semester 2024/2025:
 Timetable is not available yet
 Timetable for summer semester 2024/2025:
 Timetable is not available yet
 The course is a part of the following study plans:

 PIL bak.prez.19/20 (compulsory course)
 TUL bak.prez.19/20 (compulsory course in the program)
 PIL bak.prez.20/21 (compulsory course)
 TUL bak.prez.20/21 (compulsory course in the program)
 TUL bak.prez.21/22 (compulsory course in the program)
 DOS bak.prez.21/22 (compulsory course)
 LED bak.prez.21/22 (compulsory course)
 TUL bak.prez.22/23 (program TUL) (compulsory course in the program)
 bak.prez.od 23/24 (pro TET) (compulsory course)
 TUL bak.prez.23/24 (program TUL) (compulsory course in the program)
 bak.prez.od 24/25 (pro TET) (compulsory course)
 bak.komb.od 24/25 (pro TET) (compulsory course)
 TUL bak.prez.24/25 (program TUL) (compulsory course in the program)