Mathematics II
Content
Objective
This course deals in depth with the subject of linear algebra. Linear algebra is the basis of many techniques used in many fields such as computer science, automata and economics. Throughout the course, the basic concepts of linear algebra will be explored with an emphasis on real Euclidean spaces and vector spaces of polynomials.
In this context, the objectives of the course are:
- Introduce students to all the axiomatic definitions and signs of linear algebra: group, vector space, matrix ...
- Teach students a number of simple computational techniques that will facilitate solving linear algebra problems: solving a linear system, factoring a polynomial, simplifying a rational fraction, inverting a matrix.
- Explain the concept of dimension and its properties in a vector space.
- Show students the link between a linear function and its different matrix representations.
Course Content
1. Geometry of the plane and the space: Collinearity / orthogonality of the vectors of R ^ 2 or R ^ 3.
2. Geometry of the plane and the space: Application to the study to the study of the lines of the plane / of the lines and the planes of space
3. Linear systems: Gaus pivot method for solving linear systems. Geometric interpretation for systems with 2 or 3 unknowns. Discussion of the solutions of a system with parameters
4. Matrices: Definition and properties of operations on matrices. Matrix writing of a linear system. Reversible matirces. Linear application associated with a matrix.
5. Complex numbers: Cartesian and polar representation of a complex. Application to geometry and trigonometry
6. Complex numbers: Equation of degree 2 with complex coefficients. Nth roots of a complex.
7. Polynomials: Operations on polynomials. Euclidean division Roots of a polynomial
8. Partial examination / Arasinav
9. Polynomials: Taylor formulas. Factoring on C and on R
10. Vector spaces: Definition, examples and properties. Vector subspace of a vector space.
11. Vector spaces: Free families, generating families and bases of a vector space.
12. Vector Spaces: Dimensional theory.
13. Linear applications: Definition and properties. Matrix representation of a linear application.
14. Linear applications: Kernel and image of a linear application. Rank theorem. Change of bases.
Course Learning Outcomes
The student who successfully completes this course will have skills in the following subjects:
1.solve a system of linear equations with the Gauss method and geometrically interpret the set of solutions,
2.use Euclidean geometry in dimension 2 or 3 to solve a geometry problem,
3.use complex numbers and their geometric representation to factorize a polynomial,
4.Factorize a polynomial irreducibly or simplify a rational fraction,
5.prove that a set is a vector space and determine its dimension,
6.determine if any subspaces of a given vector space are supplementary,
7. prove that an application is linear and write its matrix in given bases,
11. find the kernel and the image of a given linear function,
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
Weekly Contents
| Week | Weekly Contents |
|---|---|
| 1 | Factorial, permutation, combinaison. Concept of determinant, properties and computation of a determinant. |
| 2 | Definition of a matrix, properties, basic operations with matrix. |
| 3 | Linear equations, solving methods and interpretations of the solutions. |
| 4 | Eigenvalues and eigenvectors of a square matrix. Diagonalization of a square matrix and quadratic forms. |
| 5 | Several real variables functions – Introduction, domain, limits and continuity. |
| 6 | Partial derivatives and differantial of a functions of multiple real variables. |
| 7 | Exam 1. Partial derivatives of parametric and implicite functions. |
| 8 | Homogenous functions and Euler formula. Directional derlvative, gradient and contour lines. |
| 9 | Second order partial derivative, Schwarz theorem. |
| 10 | n.th order partial derivative Taylor and Mc-Laurin formulas. |
| 11 | Free extrema of two variables functions. |
| 12 | Exam 2. Extrema sub consraints of two variables functions. |
| 13 | Extrema sub consraints of two variables functions. Free extrema of multivariate functions. |
| 14 | Free extrema of multivariate functions. Extrema sub consraints of multivariate functions. |
Assessment System
| Activities | Number | Contribution |
|---|---|---|
| Contribution of in-term studies to overall grade | 3 | 60 |
| Contribution of final exam to overall grade | 1 | 40 |
| Total | 4 | 100 |
| Activities | Number | Contribution |
|---|---|---|
| Assignments | 1 | 10 |
| Presentation | 1 | 20 |
| Midterm Examinations (including preparation) | 1 | 30 |
| 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 |
| Make-up | 0 | 0 |
| Total | 3 | 60 |
Relation of Proficiency
| 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 | X | ||||
| 10 | Awareness of professional and ethical responsibility | X | ||||
| 11 | Knowledge of the concepts of professional life as "project management", "risk management" and "management of change" | X | ||||
| 12 | Knowledge on entrepreneurship, innovation and sustainability | X | ||||
ECTS
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Class Hours | 28 | 2 | 56 |
| Working Hours out of Class | 16 | 2 | 32 |
| Assignments | 0 | 0 | 0 |
| Presentation | 0 | 0 | 0 |
| Midterm Examinations (including preparation) | 2 | 10 | 20 |
| Project | 0 | 0 | 0 |
| Laboratory | 0 | 0 | 0 |
| Other Applications | 0 | 0 | 0 |
| Final Examinations (including preparation) | 1 | 33 | 33 |
| Quiz | 0 | 0 | 0 |
| Term Paper/ Project | 0 | 0 | 0 |
| Portfolio Study | 0 | 0 | 0 |
| Reports | 0 | 0 | 0 |
| Learning Diary | 0 | 0 | 0 |
| Thesis/ Project | 0 | 0 | 0 |
| Seminar | 0 | 0 | 0 |
| Other | 0 | 0 | 0 |
| Make-up | 0 | 0 | 0 |
| Total Workload | 141 | ||
| Total Workload / 25 | 5.64 | ||
| Credits ECTS | 6 | ||