Linear Algebra I

• 39 hrs lectures
• 26 hrs exercises

The course aims to acquaint the students with the basics of algebraic operations, linear algebra, vector and Euclidean spaces, and analytic geometry. This will enable them to attend further mathematical and engineering courses and deal with engineering problems. Another goal of the course is to develop the students' logical
thinking.
Students will be made familiar with algebraic operations,linear algebra, vector and Euclidean spaces, and analytic geometry. They will be able to work with matrix operations, solve systems of linear equations and apply the methods of linear algebra to analytic geometry and engineering tasks. When completing the course, the students will be prepared for further study of mathematical and technical disciplines.

Students are expected to have basic knowledge of secondary school mathematics.

• Karásek, J., Skula, L.: Algebra a geometrie, Cerm 2002.
• Nedoma, J.: Matematika I., Cerm 2001.
• Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR 1998.
• Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997.
• Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996.
• Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty, SNTL 1992.
• Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita 1991.
• Procházka, L. a spol.: Algebra, Academia 1990.

• Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addison Wesley 2003.
• Howard, A. A.: Elementary Linear Algebra, Wiley 2002.
• Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus 1995.
• Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982.

1. Number sets, fields. Vector spaces, subspaces, homomorphisms.
2.  Linear dependence of vectors, basis and dimension.
3. Matrices and determinants.
4. Systems of linear equations.
5. The characteristic polynomial, eigen values, eigen vectors. 
6. Jordan normal form.
7. Euclidean and unitary vector spaces.
8. Dual vector spaces. Linear forms.
9. Bilinear and quadratic forms.
10. Orthogonality. Gram-Schmidt process.
11. Inner, exterior and cross products – relations and applications.
12. Affine and euclidean spaces. Geometry of linear objects.
13. Reserve

Week 1: Fundamental notions such as vectors, matrices and operations.
Following weeks: Seminar related to the topic of the lecture given in the previous week.

Seminars with computer support are organized according to current needs. They enables students to solve algorithmizable problems by computer algebra systems.

Course-unit credit requirements: Active attendance at the seminars, at least 50% of points in written tests. There is one alternative date to correct these tests.
Form of examinations: The exam is written and has two parts.
The exercises part takes 100 minutes and 6 exercises are given to solve.
The theoretical part takes 20 minutes and 6 questions are asked.
At least 50% of the correct results must be obtained from each part. If less is met in one of the parts, then the classification is F (failed).
Exercises are evaluated by 3 points, questions by 1 point.
If 50% of each part is met, the total classification is given by the sum.
A (excellent): 22 - 24 points
B (very good): 20 - 21 points
C (good): 17 - 19 points
D (satisfactory): 15 - 16 points
E (enough): 12 - 14 points
F (failed): 0 - 11 points

Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is in the competence of the teacher.