ECTS
6 credits
Component
Faculty of Science
Description
This follows on from S1 (Algebra I), which introduced linear algebra in R², R³ and Rn, matrix calculus and polynomials with real coefficients.
The aim is to introduce a few elementary concepts of algebraic structure, and deepen work on vector spaces and linear applications, as well as polynomials.
Objectives
- Algebra structures
- Law of internal composition on a set
- Notion of associativity, commutativity, neutral element, inverse
- Notion of group, ring and body
- Calculating in a ring. Remarkable identities and binomial formula.
- Examples(C is a body, roots of unity, permutation group, ring of polynomials and endomorphisms/matrices, group of invertible automorphisms/matrices and subgroup of isometries, etc.).
- The vector space structure
- Structure of a vector space over a K body. Rn and Cn cases, space of real sequences, space of numerical functions
- Linear combinations and collinearity
- Vector subspace, vector subspace generated by a part Generating families, free families, bases, dimension, incomplete base and exchange theorem
- Sum and direct sum of subspaces, supplementary.
- Rank of a family of vectors
- Grassmann formula
- Linear applications
- Core and image
- Correspondence between linear application and matrix with all the usual properties.
- Change of base
- Invariance of the trace by change of basis and definition of the trace of an endomorphism, tr(uv)=tr(vu).
- Isomorphism and reciprocal linear application. Groups GL(E) and GL(n).
- Projection, symmetry, homothety
- Rank of a linear application, rank of a matrix. Rank theorem. Invariance of rank by composition/multiplication by invertible matrices.
- Reduced step form of a matrix, elementary operations
- Back to linear systems, link between the rank of a matrix and the number of pivots of its reduced staggered form, kernel dimension/number of free variables
- Polynomials
- Back to K[X] as a vector space
- Case ofKn[X]: change of bases, decomposition of polynomials in bases of type 1,X-a,(X-a)2...
- Proof of a root of P if there exists Q such that P=(X-a)Q
- Taylor formula, characterization of root multiplicity
- Lagrange interpolator polynomials
- Substitution of indeterminate
Teaching hours
- Algebra II, vector spaces and linear applications - CMLecture30h
- Algebra II, vector spaces and linear applications - TDTutorial30h
Necessary prerequisites
S1 mathematics program, in particular Algebra I, Geometry in the plane and complex plane, and Reasoning and set theory.
Recommended prerequisites :
S1 mathematics program.
Further information
Hourly volumes :
CM: 30 h
TD: 30 h
TP: 0
Land: 0