| Course title | Algebra 4 |
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| Course code | KMT/AL4@ |
| Organizational form of instruction | Lecture + Exercise |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 3 |
| Language of instruction | Czech |
| Status of course | Compulsory |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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Syllabus of the course Algebra 4 1. Matrices - Matrices and their basic properties - Operations with matrices - Regular matrices - Inverse matrices - Matrix rank 2. Systems of Linear Equations - Matrix representation of systems of linear equations - Gaussian elimination - Gauss-Jordan elimination - Frobenius theorem on solvability of systems of linear equations - Structure of the solution set 3. Determinants - Definition of the determinant - Properties of determinants - Computation of determinants - Laplace expansion - Cramer's rule 4. Vector Spaces - Definition of a vector space - Subspaces - Linear spans - Linear independence and dependence - Basis of a vector space - Dimension of a vector space 5. Linear Mappings - Definition of a linear mapping - Kernel of a linear mapping - Image of a linear mapping - Matrix of a linear mapping - Properties of linear mappings 6. Euclidean Vector Spaces - Inner product - Vector norm - Orthogonality - Orthogonal basis - Gram-Schmidt orthogonalization Basic Literature: Blažek, J. a kol.: Algebra a teoretická aritmetika I. SPN, Praha, 1983. Blažek, J. a kol.: Algebra a teoretická aritmetika II. SPN, Praha, 1985.
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| Learning activities and teaching methods |
| Lecture |
| Learning outcomes |
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The course aims to introduce students to the fundamental concepts and methods of linear algebra. Emphasis is placed on understanding the relationships between matrices, systems of linear equations, and vector spaces. Students will master techniques for solving algebraic problems, ranging from elementary row operations to advanced studies of the properties of finite-dimensional Euclidean spaces.
Upon successful completion of the course, the graduate will be able to: - Work with matrices: Define matrices, perform matrix operations, determine matrix rank, and compute inverse matrices. - Solve systems of linear equations: Apply Gaussian and Gauss-Jordan elimination methods, determine solvability conditions, and analyze the structure of solution sets. - Work with determinants: Compute determinants, apply their properties, and use Laplace expansion and Cramer's rule. - Understand the theory of vector spaces: Define vector spaces and subspaces, work with linear independence, bases, and dimension. - Work with linear mappings: Define linear mappings, determine their kernels and images, and analyze their basic properties. - Work in Euclidean vector spaces: Use inner products, work with orthogonality, and construct orthogonal bases. - Apply methods of linear algebra to mathematical problems and understand their significance for other areas of mathematics. |
| Prerequisites |
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Algebra 3.
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| Assessment methods and criteria |
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Mark
During the oral examination, students are required to demonstrate an active knowledge of the definitions and theorems contained in the relevant chapters of the primary course literature: - Blažek, J. et al.: Algebra and Theoretical Arithmetic I. SPN, Prague, 1983. - Blažek, J. et al.: Algebra and Theoretical Arithmetic II. SPN, Prague, 1985. The examination also includes solving a randomly selected exercise chosen from those provided at the end of the individual chapters covered during the course in the primary literature. To successfully pass the examination, it is therefore highly advisable to have solved all of these exercises in advance, either through independent study or within the course seminars and exercise sessions. |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB23) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB25) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB19) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB25) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB26) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB26) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Summer |