Linear Algebra

Garant předmětu:

Lecturer:

Tutor:

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:

1. Basic knowledge of algebra and arithmetic (secondary schools level).

2. Active participation in lectures and exercices.

3. Examination: The written part of the examination consists of nine tasks: six calculation examples and three theoretical questions. The oral part of the examination examines deeper understanding of numerical algorithms, knowledge of terms and phrases.

For more information see the page:

https://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra%20-%20anglicky/Linear%20algebra.html

Syllabus of lectures:

1.-2. 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).

3.-4. 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).

5. 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.

6.-7.Determinants, methods of calculation, Laplace expansion, calculation of inverse ma-trix, Cramer’s rule. Determinants and their applications in algebra and geometry. Dot product, area and volume.

8.-10. Similarity of matrices, eigenvalues, eigenvectors, eigenspace, generalized eigenvectors, Jordan block matrix, Jordan canonical form, transformations.

11.-12. Quadratic forms, analytic expression, polar expression, polar basis, normal expression, canonical basis, classification of quadratic forms, methods of classification, signature of quadratic forms, Sylvester’s rule.

Syllabus of tutorials:

1. Vector spaces (linear combinations, linear independence, dimension, basis, coordinates).

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2. Matrices and operations.

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3. Systems of linear equations and their solvability.

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4. Determinants and their applications. Scalar product.

5. Similarity of matrices (eigenvalues and eigenvectors).

6. Quadratic forms and their classification.

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 Light, 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.

https://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra%20-%20anglicky/Linear%20algebra.html

Lectures and exams on-site, consultations by e-mail or on-site consultation. All study materials together with a set of 600 examples are publicly available on the web.

Note:

Further information:

http://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra.html (český jazyk), https://www.fd.cvut.cz/personal/becvamar/Linearni%20algebra%20-%20anglicky/Linear%20algebra.html (anglický jazyk)

No time-table has been prepared for this course

The course is a part of the following study plans:

• Prospectus-Bachelor (elective specialized course)