Irena Penev

Linear Algebra 1 - NMAI057 (winter 2023)

Lecture:
  • Monday 10:40-12:20, S9

Tutorials:

  • Wednesday 9:00-10:30, S10 (Irena Penev)
  • Wednesday 14:00-15:30, S11 (Irena Penev)
  • Thursday 14:00-15:30, S11 (Denys Bulavka)

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



Course matrials:
Lecture Notes (so far, only Chapters 0-5 have been posted)



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 contain primarily problems similar to tutorial problems (including HW and quiz problems).

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

  1. obtain at least 50% on weekly/biweekly HW assignments (the lowest HW score will be dropped);
  2. one of the following:
    • obtain at least 70% on weekly/biweekly quizzes (the lowest quiz score will be dropped),
    • obtain at least 50% on weekly/biweekly quizzes (the lowest quiz score will be dropped) and at least 70% at the end-of-semester test (the problems on the test will be similar to quiz problems).



Lectures
Lecture 0: Mathematical induction. Modular arithmetic. Arithmetic in ℤn (slides)

Lecture 1: Systems of linear equations (slides)

Lecture 2: Matrix-vector equations. The rank of a matrix (slides)

Lecture 3: Matrix multiplication and transpose. Solving matrix equations of the form AX = B and XA = B (slides)

Lecture 4: Linear functions (slides)

Lecture 5: Invertible matrices (slides)

Lecture 6: Groups (slides)

Lecture 7: Permutations and the symmetric group. Fields (slides)

Lecture 8: Vector spaces (part I) (slides)

Lecture 9: Vector spaces (part II) (slides)

Lecture 10: Linear functions (part I) (slides)

Lecture 11: Linear functions (part II) (slides)

Lecture 12: Affine subspaces and affine functions (slides)


Tutorials
Tutorial 1

Tutorial 2

Tutorial 3

Tutorial 5

Tutorial 6

Tutorial 7

Tutorial 8

Tutorial 9


HW
HW should be submitted via the Postal Owl. You should have received the token by e-mail (if you haven't, please contact me).

HW 1 (due Friday, October 13, 2023, at noon)

HW 2 (due Friday, October 27, 2023, at noon)

HW 3 (due Friday, November 10, 2023, at noon)

HW 4 (due Friday, November 17, 2023, at noon) - statement of Problem 4 corrected on Monday, November 13 at 4:25 pm.

HW 5 (due Friday, December 1, 2023, at noon)

HW 6 (due Friday, December 8, 2023, at noon)

HW 7 (due Friday, December 22 December 29, 2023, at noon)

HW 8 (due Friday, January 5, 2024, at noon)

HW 9 (due Friday, January 12, 2024, at noon)

There will be no quiz on January 10-11.




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