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