Linear Algebra(ING207)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| ING207 | Linear Algebra | 3 | 2 | 2 | 0 | 3 | 5 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | French |
| Course Type | Compulsory |
| Course Level | Bachelor Degree |
| Course Instructor(s) | Marie Christine PEROUEME mcperoueme@voila.fr (Email) |
| Assistant | |
| Objective |
Mathematical problems such as solving systems linear differentials (which occur in many areas physics such as mechanics or electronics) or analysis in principal components in statistics use the diagonalization of square matrices. Determine if a matrix is ??diagonalizable, and in in this case, diagonalizing it is therefore the key to this course. In this context, the objectives of this course are: • Explain to students how the determinant of a matrix is defined using permutations and their signature, in particular in order to be able to define the characteristic polynomial. • Teach students to determine the specific elements of a matrix. • Demonstrate to the students the conditions of diagonalization of a matrix. • Explain to the students how to use diagonalization to solve linear systems. |
| Content |
1. Symmetric group: decomposition into products and signature of a permutation 2. Determinants: definition, properties and calculation rules 3. Determinants: determinants of "small" dimensions, classical determinants 4. Diagonalization: Introduction and first examples 5. Classical determinant applications 6. Diagonalization: criterion of diagonalization (case of multiple eigenvalues) 7. Diagonalization: diagonalization of "small" dimension matrices 8. Partial Examination 9. Diagonalization: calculation of the nth powers of a diagonalizable matrix 10. Polynomials of matrices, canceling polynomials - Cayleigh Hamilton Theorem 11. Application to the calculation of the nth powers of a matrix (diagonalizable or not) 12. Application to linear recurrent sequences 13. Application to differential systems (diagonalizable case) 14. Practical studies |
| Course Learning Outcomes |
The student who will take this course will develop the elements of competence following and will be able to: 1. Calculate the decomposition in cycles with disjoint supports and signing a permutation. 2. Calculate the determinant of a square matrix. 3. Determine the characteristic polynomial (and therefore, the eigenvalues) of a matrix. 4. Determine the eigenspaces of a matrix. 5. Illustrate on geometric examples (homothety, rotation, symmetry ...) the dimension and direction of the proper spaces. 6. Prove if a matrix is diagonalizable in R or in C. 7. Determine the diagonalized matrix as well as the associated matrix passage. 8. Solve linear systems (equations differential or recurrent sequences). |
| Teaching and Learning Methods | Lectures and supervised works/tutorials |
| References |
1. Lectures notes ans worksheets 2. http://braise.univ-rennes1.fr/braise.cgi 3. http://www.unisciel.fr |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | The grup of permutations. |
| 2 | Decomposition into disjoint cycles, decomposition into transposition and signature of a permutation. |
| 3 | Determinant : definition and basic propoerties |
| 4 | Some methods to compute determinant |
| 5 | Some examples of classic determinants. |
| 6 | eigenvalues of a determinant and some geometric examples. |
| 7 | Characteristic polynomial, eigenvalues and eigenvectors |
| 8 | Diagonalizable matrixs |
| 9 | Midterm exam |
| 10 | The Cayley–Hamilton theorem |
| 11 | Different methods for computing the powers of a matrix. |
| 12 | Linear recurrence sequences of order 2 or 3. |
| 13 | Systems of homogeneous linear differential equations with constant coefficients. |
| 14 | Systems of nonhomogeneous linear differential equations with constant coefficients. |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 0 | 0 |
| Contribution of final exam to overall grade | 0 | 0 |
| Toplam | 0 | 0 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 0 | 0 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 1 | 40 |
| Project | 0 | 0 |
| Laboratory | 0 | 0 |
| Other Applications | 0 | 0 |
| Quiz | 0 | 0 |
| Term Paper/ Project | 0 | 0 |
| Portfolio Study | 0 | 0 |
| Reports | 0 | 0 |
| Learning Diary | 0 | 0 |
| Thesis/ Project | 0 | 0 |
| Seminar | 0 | 0 |
| Other | 0 | 0 |
| Toplam | 1 | 40 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Knowledge and understanding of a wide range of basic sciences (math, physics, ...) and the main concepts of engineering | X | ||||
| 2 | Ability to combine the knowledge and skills to solve engineering problems and provide reliable solutions | X | ||||
| 3 | Ability to select and apply methods of analysis and modeling to ask, reformulate and solve the complex problems of industrial engineering | X | ||||
| 4 | Ability to conceptualize complex systems, processes or products under practical constraints to improve their performance, ability to use innovative methods of design | X | ||||
| 5 | Ability to design, select and apply methods and tools needed to solve problems related to the practice of industrial engineering, ability to use computer technology | X | ||||
| 6 | Ability to design experiments, collect and interpret data and analyze results | X | ||||
| 7 | Ability to work independently, ability to participate in working groups and have a multidisciplinary team spirit | X | ||||
| 8 | Ability to communicate effectively, ability to speak at least two foreign languages | X | ||||
| 9 | Awareness of the need for continuous improvement of lifelong learning, ability to keep abreast of scientific and technological developments to use the tools of information management | |||||
| 10 | Awareness of professional and ethical responsibility | |||||
| 11 | Knowledge of the concepts of professional life as "project management", "risk management" and "management of change" | |||||
| 12 | Knowledge on entrepreneurship, innovation and sustainability | |||||
| 13 | Understanding of the effects of Industrial Engineering applications on global and social health, environment and safety. | |||||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Class Hours | 14 | 4 | 56 |
| Working Hours out of Class | 14 | 2 | 28 |
| Midterm Examinations (including preparation) | 2 | 15 | 30 |
| Final Examinations (including preparation) | 1 | 12 | 12 |
| Total Workload | 126 | ||
| Total Workload / 25 | 5.04 | ||
| Credits ECTS | 5 | ||