Irena Penev

Linear Algebra 2 - NMAI058 (summer 2025)

Lecture:
  • Wednesday 10:40-12:10, N1

Tutorials:

  • Tuesday 10:40-12:10, N5 N3 [note the room change]
  • Tuesday 12:20-13:50, N5 N3 [note the room change]
  • Tuesday 14:00-15:30, N7

Office hourse: Right after lecture on Wednesdays (in the IMPAKT corridor), or by appointment.

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



Course matrials:
Lecture Notes



Course content
Inner product spaces:
  • norm induced by an inner product
  • Pythagoras theorem, Cauchy-Schwarz inequality, triangle inequality
  • orthogonal and orthonormal system of vectors, Fourier coefficients, Gram-Schmidt
  • orthogonalization
  • orthogonal complement, orthogonal projection
  • the least squares method
  • orthogonal matrices
Determinants:
  • basic properties
  • Laplace expansion of a determinant, Cramer's rule
  • adjugate matrix
  • geometric interpretation of determinants
Eigenvalues and eigenvectors:
  • basic properties, characteristic polynomial
  • Cayley-Hamilton theorem
  • similarity and diagonalization of matrices, spectral decomposition, Jordan normal form
  • symmetric matrices and their spectral decomposition
  • (optionally) companion matrix, estimation and computation of eigenvalues: Gershgorin discs and power method
Positive semidefinite and positive definite matrices:
  • characterization and properties
  • methods: recurrence formula, Cholesky decomposition, Gaussian elimination, Sylvester's criterion
  • relation to inner products
Bilinear and quadratic forms:
  • forms and their matrices, change of a basis
  • Sylvester's law of inertia, diagonalization, polar basis
Topics on expansion (optionally):
  • eigenvalues of nonnegative matrices
  • matrix decompositions: Householder transformation, QR, SVD, Moore-Penrose pseudoinverse of a matrix
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:

  1. obtain at least 60% on weekly/biweekly HW assignments (the lowest HW score will be dropped);
  2. 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 and the end-of-semester test will each be 90 minutes long.




Lectures
Lectures 0-11 (from Linear Algebra 1) can be found here.

Lecture slides will be updated as the semester proceeds. Slides are closely based on the Lecture Notes. I recommend using the Lecture Notes (rather than slides) for studying at home.

Lecture 12: Matrices of linear functions between non-trivial, finite-dimensional vector spaces (slides) [covers section 4.5 of the Lecture Notes]

Lecture 13: Complex numbers. Scalar (inner) products (slides) [covers sections 0.3 and 6.1 of the Lecture Notes]

Lecture 14: The norm. Orthogonal and orthonormal bases (slides) [covers section 6.2 and subsections 6.3.1-6.3.3 of the Lecture Notes]

Lecture 15: Gram-Schmidt orthogonalization (slides) [covers subsection 6.3.4 of the Lecture Notes]

Lecture 16: The orthogonal complement of a subspace. Orthogonal projection onto a subspace (slides) [covers sections 6.4 and 6.5 of the Lecture Notes]

Lecture 17: Orthogonal projection onto subspaces of ℝn. The method of least squares (slides) [covers sections 6.6 and 6.7 of the Lecture Notes]

Lecture 18: Permutation matrices. Orthogonal matrices (slides) [covers subsection 2.3.7 and sections 6.8 and 6.9 of the Lecture Notes]


Tutorials
Tutorial 1

Tutorial 2

Tutorial 3

Tutorial 4


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, March 7, 2025, at noon)

HW 2 (due Friday, March 14, 2025, at noon)

HW 3 (due Friday, March 28, 2025, at noon)

HW 4 (due Friday, April 4, 2025, at noon)




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