# Karel Král

kralka (AT) iuuk.mff.cuni.cz

# Linear Algebra II

## Solved exercises

I will try to add solutions to the exercises that are going to appear during our classes. Please report any errors or typos you will find. If something is not clear ask me during our classes or by email. This project is new so there are not going to be all solutions. If you want me to add some particular solution or at least a hint please let me know by email.

## Classes, homework

### May 24

Quadratic forms: problems from sbirka.

Applications of linear algebra.

Linear algebra III (contact Milan Hladík).

Linear algebra in combinatorics book by prof Matoušek. There is a course from time to time with a similar structure as the book.

You will meet a lot of other courses in computer graphics, numerical mathematics, data signal processing, optimization, complexity theory, etc. where you will use linear algebra.

The additional homework is online: hw.pdf.

### May 17

Theory recap, sample exam test.

### May 10

Cholesky decomposition, testing if a matrix is positive definite.

### May 3

Finishing Jordan normal form.

### Apr 24

By Peter Korcsok.

Fifth test.

Pdf with problems.

### Apr 17

Eigenvalues, eigenvectors, algebraic vs geometric multiplicity. Using Laplace expansion to compute characteristic polynomials of bigger matrices. Determining if a matrix is diagonalizable and finding the similar diagonal matrix and the corresponding change of basis.

We discussed Markov Chains and Markov Chain Monte Carlo where we are sampling to determine properties of the studied stochastic process (uses in biology...). There are whole books on these topics, use your search engine or visit a library.

I mentioned Spectral Graph Theory where we connect graph properties to eigenvalues of incidence or adjacence matrix. People tryed to use this even to study human brain and seisures, see for instance Niso etal., Tharp, Gersch, Direito etal.

### Apr 12

Fourth test.

Eigenvalues, eigenvectors, algebraic vs geometric multiplicity.

### Apr 5

Finished determinants (inverse matrix using adjugate matrix, counting the number of spanning trees in an arbitrary given graph), problems 1, 2, 3, 8 from pdf. Next time we will do eigenvectors and eigenvalues, but there are solutions written for all problems in the pdf.

### Mar 29

Third test.

Determinants -- definition, Gaussian elimination with geometric intuition in the real plane, Laplace expansion. Problems from pdf.

### Mar 22

Linear maps, all problems from pdf.

### Mar 15

Second test. Least squares method. Linear maps (also known as linear transformations). Programming assignment (page 2 of the pdf).

### Mar 8

Problems 1, 2, 3, 4 from pdf.

### Mar 1

First test. Dot product, Gram-Schmidt ortonormalisation algorithm and example. Problems 1, 2, 4, 5 from pdf.

### Feb 22

Shamir's Secret Sharing as a motivation for finite fields and linear algebra over those.

Change of basis and its matrix. Dot product, motivation, use, definition. Problems 1, 2, 3, 4, 5 from pdf.

When we need complex numbers we will do an intro to what we will be using. If you are interested in self-studying check the awesome video by Prof. Strang (53 minutes). To watch this video you need just some basic trigonometry. The video introduces the important aspects of complex numbers. Pay attention especially to the $$r e^{i\Theta}$$ form of complex numbers and the multiplication property of adding angles. The video also derives the formula $$e^{i \pi} = -1$$ and answers the question what is $$i^i$$. There is also a mention of the Mandelbrot set. Beware that electrical engineers often use $$j$$ instead of $$i$$.

## Point results

I publish the results only under a nickname.

Table with acquired points.

## Credit requirements

Copy from the SIS:

For passing the class you should obtain at least 70% of total marks from the tests given during the semester and a practical programming homework. Students can obtain addidional marks by homework (the amount of homework depends on missing marks). Homework will be given at the end of the semester if someone needs it.