Irena Penev
Linear Algebra 1 - NMAI057 (winter 2024)
Lecture:
- Wednesday 10:40-12:10, S9
Tutorials:
- Thursday 12:20-13:50,
N9 N6 [note the room change]
- Thursday 14:00-15:30, N9
- Thursday 15:40-17:10,
N9 N7 [note the room change]
Office hourse: Right after lecture on Wednesdays (or by appointment).
Contact (by e-mail): ipenev [at] iuuk [dot] mff [dot] cuni [dot] cz
Course matrials:
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 HW and tutorial 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:
- obtain at least 60% on weekly/biweekly HW assignments (the lowest HW score will be dropped);
- score at least 70% on the end-of-semester test.
HW will contain "exercises" and "problems." Exercises will be routine computations, and the end-of-semester test will consist of some subset of those exercises (with numbers changed). Problems will be either more complex computations or proofs.
- Students may discuss HW with each other in order to exchange ideas, but they are required to write up solutions by themselves. If two or more students submit (nearly) identical solutions to HW exercises/problems that are not completely routine, they will get a zero for the exercise(s)/problem(s) in question. The same applies to solutions obtained from the Internet (e.g. generated by ChatGPT). Repeat violations may lead to disciplinary action.
- Note: A "completely routine exercise" might be something like "multiply these two matrices." In such cases, two students working independently might indeed be expected to produce identical solutions. However, proofs are definitely not "completely routine exercises."
- HW must be submitted on time. Extensions are possible only with a note from a doctor or from the university (i.e. the Student Affairs Department).
- Some students will arrive in Prague after the start of the semester (due to visa delays or other reasons). Such students should use the Lecture Notes to study, and they must submit HW on time. HW can be submitted online, which means that presence in Prague is not necessary, and it will not count as a justification for submitting HW late.
Both the end-of-semester test and the final exam will be closed book. Students will not be allowed to leave the room during the test/exam (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).
|
Lectures
|
Lecture 0: Mathematical induction. Modular arithmetic. Arithmetic in ℤn (slides) [covers sections 0.1 and 0.2 of the Lecture Notes]
Lecture 1: Systems of linear equations (slides) [covers sections 1.1, 1.2, and 1.3 of the Lecture Notes; subsection 1.3.7 is optional reading]
Lecture 2: Matrix-vector equations. The rank of a matrix. Matrix operations (slides) [covers sections 1.4-1.8 of the Lecture Notes]
Lecture 3: A first look at linear functions and their matrices (slides) [covers section 1.10 of the Lecture Notes]
Lecture 4: Invertible matrices (slides) [covers section 1.11 of the Lecture Notes]
Lecture 5: Groups (slides)
Lecture 6: Permutations and the symmetric group. Fields (slides)
|
Tutorials
HW
|
HW should be submitted via the Postal Owl. You will receive the token when the first HW assignment is posted.
HW 1 (due Friday, October 11, 2024, at noon)
HW 2 (due Friday, October 18, 2024, at noon)
HW 3 (due Friday, November 1, 2024, at noon)
HW 4 (due Friday, November 8, 2024, at noon)
HW 5 (due Friday, November 22, 2024, at noon)
|
|