Irena Penev

Linear Algebra 2 - NMAI058 (summer 2024)

Lecture:
  • Wednesday 12:20-13:50, S9

Tutorials:

  • Monday 10:40-12:10, S11 (Irena Penev) [canceled]
  • Monday 12:20-13:50, S11 (Todor Antić)
  • Tuesday 14:00-15:30, S11 (Irena Penev)

Tutorials on February 19 and May 6 will be turned into lectures.


Contact (by e-mail):
  • ipenev [at] iuuk [dot] mff [dot] cuni [dot] cz    (Irena Penev)
  • todor[at] kam [dot] mff [dot] cuni [dot] cz    (Todor Antić)

Course matrials:
Lecture Notes (so far, only chapters 0-8 have been posted)



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. [UPDATE: Three exams have been scheduled in SIS.]

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.

[UPDATE: The end-of-semester test has been scheduled for Wednesday, May 29, at noon in S9. The test will take 90 minutes. If you are unable to take the test on that day, or if you take it and fail, you will get a second chance (provided you have earned at least 60% on HW). The second test will be scheduled later, but in any case, it will be after the first exam.]



Lectures
Lectures 0-12 can be found on the web page of the Linear Algebra 1 course from last semester (here).

Complex Numbers (slides)

Lecture 13: Matrices of linear functions between non-trivial, finite-dimensional vector spaces (slides)

Lecture 14: Scalar (inner) products (slides)

Lecture 15: Orthogonal and orthonormal bases. Gram-Schmidt orthogonalization (slides)

Lecture 16: The orthogonal complement of a subspace. Orthogonal projection onto a subspace (slides)

Lecture 17: Permutation matrices. Orthogonal matrices (slides)

Lecture 18: Determinants (slides)

Lecture 19: Laplace expansion. Cramer's rule. The adjugate matrix (slides)

Lecture 20: Applications of determinants: volume and polynomials (slides)

Lecture 21: Eigenvalues and eigenvectors (slides)

Lecture 22: The Cayley-Hamilton theorem. Diagonalization (slides)


Tutorials
Tutorial 1

Tutorial 2

Tutorial 3

Tutorial 4

Tutorial 5

Tutorial 6

Tutorial 7

Tutorial 8

Tutorial 9


HW
HW should be submitted via the Postal Owl. You will receive the token by e-mail.

HW 1 (due Monday, March 4, 2024, at 10 am)

HW 2 (due Monday, March 11, 2024, at 10 am)

HW 3 (due Monday, March 18, 2024, at 10 am)

HW 4 (due Monday, April 8, 2024, at 10 am)

HW 5 (due Monday, April 22, 2024, at 10 am)

HW 6 (due Monday, April 29, 2024, at 10 am)

HW 7 (due Monday, May 6, 2024, at 10 am)




home