Algebra II, vector spaces and linear applications

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.

- 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 of_Kn[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

S1 mathematics program, in particular Algebra I, Geometry in the plane and complex plane, and Reasoning and set theory.

Recommended prerequisites :

S1 mathematics program.

Hourly volumes :

            CM: 30 h

            TD: 30 h

            TP: 0

            Land: 0