SubjectsSubjects(version: 877)
Course, academic year 2020/2021
  
Linear Algebra 1 - NMAI057
Title: Lineární algebra 1
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information: https://kam.mff.cuni.cz/~balko/ln12021/LN1.html
Guarantor: doc. Mgr. Milan Hladík, Ph.D.
doc. Mgr. Petr Kolman, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Algebra
Annotation -
Last update: G_I (11.04.2003)
Basics of linear algebra (vector spaces and linear maps, solutions of linear equations, matrices).
Course completion requirements -
Last update: Mgr. Pavel Hubáček, Ph.D. (08.10.2020)

Tutorial requirements: there will a test during the time of the tutorial on October 20, 2020, November 10, 2020, December 8, 2020, and January 5, 2021. In order to get the tutorial credit, it is necessary to get at least 60% of the points from the four tests in total. In case of insufficient points from the tests, it will be possible to obtain the tutorial credit based on additional homework agreed upon with the TA at the end of the semester.

Literature -
Last update: Andrew Goodall, D.Phil. (11.10.2017)

D. Poole. Linear Algebra, A Modern Introduction. 3rd Int. Ed., Brooks Cole, 2011. Chapters 1,2,3,6.

Also useful:

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

C. D. Meyer. Matrix analysis and applied linear algebra. SIAM, Philadelphia, PA, 2000.

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

R. Beezer, A First Course in Linear Algebra - a free online textbook. http://linear.ups.edu/html/fcla.html

Teaching methods -
Last update: doc. Mgr. Milan Hladík, Ph.D. (29.09.2020)

Moodle course: https://dl1.cuni.cz/course/view.php?id=10373&lang=en

Requirements to the exam -
Last update: doc. RNDr. Jiří Fiala, Ph.D. (14.10.2019)

The exam is primarily oral. The students will be given enough time to prepare their answers in written form.

The exam is focused on theory - knowledge of concepts, facts, arguments and applications (i.e. definitions, theorems, proofs and examples).

Class credits ("zápočet") are prerequisite for taking the examination.

Syllabus -
Last update: doc. Mgr. Milan Hladík, Ph.D. (11.05.2020)

Systems of linear equations:

  • matrix form, elementary row operations, row echelon form
  • Gaussian elimination
  • Gauss-Jordan elimination

Matrices:

  • matrix operations, basic types of matrices
  • nonsingular matrix, inverse of a matrix

Algebraic structures:

  • groups, subgroups, permutations
  • fields and finite fields in particular

Vector spaces:

  • linear span, linear combination, linear dependence and independence
  • basis and its existence, coordinates
  • Steinitz' replacement theorem
  • dimension, dimensions of sum and intersection of subspaces
  • fundamental matrix subspaces (row space, column space, kernel)
  • rank-nullity theorem

Linear maps:

  • examples, image, kernel
  • injective linear maps
  • matrix representations, transition matrix, composition of linear maps
  • isomorphism of vector spaces

Topics on expansion:

  • introduction to affine spaces and relation to linear equations
  • LU decomposition
 
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