Irena Penev

Linear Algebra 1 - NMAI057 (winter 2026)

Lecture:
  • Monday 17:20-18:50, S9 (Irena Penev)

Tutorials:

  • Tuesday 10:40-12:10, N3 (Irena Penev)
  • Thursday 10:40-12:10, N7 (Morawski Adam)

Mentoring (math/theory):

  • Wednesday 17:20, S7 (Anastasia Japoshvili)

Office hours: Right after lecture on Mondays and right after the tutorial on Tuesdays (or by appointment).

Contact (by e-mail): ipenev [at] iuuk [dot] mff [dot] cuni [dot] cz



Course materials:
Lecture Notes



Course content
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
Course requirements and evaluation
Students enrolled in the lecture are required to also enroll in one of the tutorials. Tutorial credit ("zápočet") is a prerequisite for the exam.

The final exam will be written, and it will be similar to quiz/tutorial/practice exercises/problems. In rare cases, a student may be invited to take an oral exam (in addition to the written exam).

To obtain tutorial credit, students must satisfy both of the following two requirements:

  1. obtain at least 50% on weekly/biweekly quizzes (the two quiz scores will be dropped);
  2. score at least 70% on the end-of-semester test (which will contain only routine computational problems).

All assessment (quizzes, tests, exams) will be closed book: no notes, no electronic devices, no collaboration. Students will not be allowed to leave the room during quizzes/tests/exams (for example, to go to the restroom); exceptions are possible only for medical reasons, and with proper documentation from a Czech doctor or from the university. The exam will be 90 minutes long, and the end-of-semester test will be 60-90 minutes long (to be determined). A missed quiz can be made up only under extraordinary circumstances, and for this, students must provide a note from a Czech doctor or from the Student Affairs Department.




Lectures
Lecture 0: Modular arithmetic. Fermat's Little Theorem. Arithmetic in ℤn (slides) [covers sections 0.2 of the Lecture Notes]

Lecture 1: Systems of linear equations (slides) [covers sections 1.1-1.3 of the Lecture Notes]


Tutorials
Tutorial 1
There will be a quiz on October 13-15: one problem similar to Exercise 4, and one problem similar to Exercise 6 (both from Tutorial 1).





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