Irena Penev
Linear Algebra 2 - NMAI058 (summer 2023)
Lecture:
Tutorials:
- Monday 10:40-12:10, S11
- Monday 14:00-15:30, S11
- Tuesday 10:40-12:10, S10
Contact (by e-mail): ipenev [at] iuuk [dot] mff [dot] cuni [dot] cz
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
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Course requirements and evaluation
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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:
- obtain at least 50% on weekly/biweekly HW assignments (the lowest HW score will be dropped);
- 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).
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Lectures
Tutorials
HW
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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, February 24, 2023, at 10 am)
HW 2 (due Friday, March 3, 2023, at 10 am)
HW 3 (due Friday, March 17, 2023, at 10 am)
HW 4 (due Friday, March 31 April 7, 2023, at 10 am)
HW 5 (due Friday, April 14, 2023, at 10 am)
HW 6 (due Friday, April 28, 2023, at 10 am)
HW 7 (due Friday, May 12, 2023, at 10 am)
HW 8 (due Friday, May 19, 2023, at 10 am)
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