Linear Algebra 2 - NMAI058
Title: Lineární algebra 2
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: summer
E-Credits: 5
Hours per week, examination: summer 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, English
Teaching methods: full-time
Guarantor: prof. Mgr. Milan Hladík, Ph.D.
doc. Mgr. Petr Kolman, Ph.D.
Teacher(s): RNDr. Martin Černý
doc. RNDr. Jiří Fiala, Ph.D.
Mgr. Elif Garajová, Ph.D.
prof. Mgr. Milan Hladík, Ph.D.
doc. Mgr. Petr Kolman, Ph.D.
Mgr. Matyáš Lorenc
RNDr. Jana Maxová, Ph.D.
RNDr. Ondřej Pangrác, Ph.D.
Irena Penev, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Algebra
Is incompatible with: NALG086, NUMP004, NUMP003
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Annotation -
Continuation of MAI057 - special matrices, determinants, eigenvalues, examples of applications of linear algebra.
Last update: FIALA/MFF.CUNI.CZ (17.02.2010)
Course completion requirements -

The relevant information for the English section of this course in summer semester of 2025 can be found on the course web page: https://iuuk.mff.cuni.cz/~ipenev/NMAI058S2025.html

Last update: Penev Irena, Ph.D. (18.02.2025)
Literature -

W. Gareth. Linear Algebra with Applications. Jones and Bartlett Publishers, Boston, 4th edition, 2001.

C. D. Meyer. Matrix analysis and applied linear algebra. SIAM, Philadelphia, PA, 2000. http://www.matrixanalysis.com/DownloadChapters.html

G. Strang. Linear algebra and its applications. Thomson, USA, 4rd edition, 2006.

Last update: Hladík Milan, prof. Mgr., Ph.D. (22.11.2012)
Requirements to the exam -

The relevant information for the English section of this course in summer semester of 2025 can be found on the course web page: https://iuuk.mff.cuni.cz/~ipenev/NMAI058S2025.html

Last update: Penev Irena, Ph.D. (18.02.2025)
Syllabus -

Inner product spaces:

  • norm induced by an inner product
  • Pythagoras theorem, Cauchy-Schwarz inequality, triangle inequality
  • orthogonal and orthonormal system of vectors, Fourier coefficients, Gram-Schmidt orthogonalization
  • orthogonal complement, orthogonal projection
  • the least squares method
  • orthogonal matrices

Determinants:

  • basic properties
  • Laplace expansion of a determinant, Cramer's rule
  • adjugate matrix
  • geometric interpretation of determinants

Eigenvalues and eigenvectors:

  • basic properties, characteristic polynomial
  • Cayley-Hamilton theorem
  • similarity and diagonalization of matrices, spectral decomposition, Jordan normal form
  • symmetric matrices and their spectral decomposition
  • (optionally) companion matrix, estimation and computation of eigenvalues: Gershgorin discs and power method

Positive semidefinite and positive definite matrices:

  • characterization and properties
  • methods: recurrence formula, Cholesky decomposition, Gaussian elimination, Sylvester's criterion
  • relation to inner products

Bilinear and quadratic forms:

  • forms and their matrices, change of a basis
  • Sylvester's law of inertia, diagonalization, polar basis

Topics on expansion (optionally):

  • eigenvalues of nonnegative matrices
  • matrix decompositions: Householder transformation, QR, SVD, Moore-Penrose pseudoinverse of a matrix

Last update: Hladík Milan, prof. Mgr., Ph.D. (28.03.2022)