Algebra II, vector spaces and linear applications

  • 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.

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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
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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.

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Further information

Hourly volumes :

            CM: 30 h

            TD: 30 h

            TP: 0

            Land: 0

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