Irena Penev

Linear Algebra 1 - NMAI057 (winter 2022)

Lecture:
  • Wednesday 12:20-13:50, S4

Tutorials:

  • Tuesday 10:40-12:10, S4
  • Tuesday 14:00-15:30, S10
  • Wednesday 9:00-10:30, S10

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



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 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: Modular arithmetic. Arithmetic in ℤn (Lecture Notes)

Lecture 1: Systems of linear equations. Row reduction (Lecture Notes)

Lecture 2: Vectors. Matrix-vector multiplication. Matrix-vector equations (Lecture Notes)

Lecture 3: Matrix operations: addition, multiplication, and transpose (Lecture Notes)

Lecture 4: Linear transformations. Invertible matrices (Lecture Notes)

Lecture 5: Groups and permutations (Lecture Notes)

Lecture 6: Fields and vector spaces (Lecture Notes)

Lecture 7: Bases of vector spaces. The column space, row space, and null space (Lecture Notes)

Lecture 8: Linear transformations between arbitrary vector spaces (Lecture Notes)

Lecture 9: More about linear transformations. Isomorphisms (Lecture Notes)


Tutorials
Notes on induction

Tutorial 1

Tutorial 2

(Tutorial 3 was turned into a lecture.)

Tutorial 4

Notes on solving matrix equations (AX = B and XA = B)

Tutorial 5

Tutorial 6

Tutorial 7

Tutorial 8

Tutorial 9

Tutorial 10

Tutorial 11 (Some solutions)

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 Thursday, October 20, 2022, at 10 am)

HW 2 (due Thursday, October 27, 2022, at 10 am)

HW 3 (due Thursday, November 3, 2022, at 10 am)

HW 4 (due Thursday, November 17, 2022, at 10 am)

HW 5 (due Thursday, November 24, 2022, at 10 am)

HW 6 (due Thursday, December 8, 2022, at 10 am)

HW 7 (due Thursday, December 15, 2022, at 10 am)

HW 8 (due Thursday, December 22, 2022, at 10 am)

HW 9 (due Thursday, January 5, 2023, at 10 am)