**Mathematics 4B **
(101MT4B)

*The page will be updated during the progress of the 101MT4B course.*

The collection comprises solved problems that (a) illustrate the theory expounded in lectures, and (b) serve as sample exam problems.

**February 17 and February 24:** Introductory lectures
to refresh the knowledge of the elements of linear algebra. There is no
lecture presentation available; links to Wikipedia articles are given
instead.

Topics:

1) Matrix with real entries, operations (addition, scalar multiplication, matrix multiplication - does not commute), transposition, matrix rank, non-singular matrix, inverse matrix.

https://en.wikipedia.org/wiki/Matrix_(mathematics)

2) Vector space (n-tuples of real numbers, polynomials of a fixed degree), linear combination, linearly dependent and independent vectors, basis.

https://en.wikipedia.org/wiki/Vector_space

3) System of linear algebraic equations, augmented matrix, Gaussian elimination, row echelon form, the existence and uniqueness of the solution (Rank-Nullity Theorem or Rouché-Capelli Theorem or Rouché-Frobenius Theorem), null space (kernel).

https://en.wikipedia.org/wiki/System_of_linear_equations

https://en.wikipedia.org/wiki/Gaussian_elimination

https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem

4) Determinant of 2x2 and 3x3 matrices, Laplace's formula (expansion along a row or a column).

https://en.wikipedia.org/wiki/Determinant

5) Eigenvalue, eigenvector, characteristic polynomial, algebraic multiplicity, geometric multiplicity, properties of eigenvalues, spectrum of a matrix, spectral radius.

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

**February
25, March 3:** **
PDF presentation**
(complex numbers,
eigenvalues and eigenvectors, Gershgorin theorem)

Visit
**Collection of solved
problems **for
examples.

**A:** **
PDF presentation**
(vector (linear) space, normed vector space, vector norms, matrix
norms, condition number)

Visit
**Collection of solved
problems **for
examples.

**B:** **
PDF presentation**
(2nd order linear ODEs, null space, structure of solutions, methods:
identification of parameters and variation of constants)

Visit
**Collection of solved
problems **for
examples.

**C:** **
PDF presentation**
(motivation: beam stability, eigenvalue problem for the
2nd order linear ODEs, eigenvalues and
eigenfunctions, solvability of eigenproblems)

Visit
**Collection of solved
problems **for
examples.

**D****:** ** PDF presentation**
(finite difference method in 1D: approximation of
derivatives at mesh nodes, finite difference equations, boundary value
problem approximated by a linear algebra problem, approximation of
eigenvalues and eigenfunctions)

Visit
**Collection of solved
problems **for
examples.

**E****:** ** PDF presentation**
(functional of energy, variational principle,
differential operator, positive definite operator, Friedrichs inequality,
Ritz method)

Visit
**Collection of solved
problems **for
examples.

**F:** ** PDF presentation**
(finite element method in 1D problems)

Visit
**Collection of solved
problems **for
examples.

**F****inal
exam on **-------** ***(to be announced)*

Register to the KOS system, please.

- The exam will comprise several problems; written solutions and answers are expected.
- Cell (mobile) phones, laptops, textbooks, etc. are not permitted for the exam.
- Only
**non**-programmable scientific calculators can be used. However, they are unnecessary because all calculations can easily be done by hand. - Students can also use the official cheat (crib) sheet and an individual hand-written cheat sheet (A4 size, both sides); the individual cheat sheet must be handed in together with the individual written exam.
- Grading: 0 - 49 pts => F, 50 - 59 pts => E, 60 - 69 pts => D, 70 - 79 pts => C, 80 - 89 pts => B, 90 - 100 pts => A

I hope that all the students will pass the exam on -------. If not, the date of the next exam will be specified.

**There are many free linear algebra textbooks or courses**
**available on the Internet:**

http://linear.ups.edu/ (both online and PDF versions available)

https://www.math.ucdavis.edu/~linear/linear-guest.pdf

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/

**etc.**

*Remark: The content of the referred
Wikipedia pages is richer than the content of the lectures.*

[Last modification: Feb. 17, 2020; J. C.]