Mathematics 4B (101MT4B)

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

 

Collection of solved problems   

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.

 

 

Final exam on ------- (to be announced)
Register to the KOS system, please.

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.]