Algebra II, vector spaces and linear applications

This course is a continuation of the S1 course (Algebra I) where linear algebra in R², R³ and ^Rn, matrix calculus and polynomials with real coefficients were introduced.

The objective is to introduce some elementary concepts of algebraic structure, and to deepen the work on vector spaces and linear applications, as well as polynomials.

- The structures in algebra

• Law of internal composition on a set
• Notion of associativity, commutativity, neutral element, inverse
• Notion of group, ring and body
• Calculus in a ring. Remarkable identities and binomial formula.
• Examples(C is a body, roots of unity, group of permutations, ring of polynomials and endomorphisms/matrices, group of invertible automorphisms/matrices and subgroup of isometries, etc.)

- The structure of a vector space

• Structure of a vector space over a body K. Cases ^Rn and ^Cn, 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 basis and exchange theorem
• Sum and direct sum of subspaces, supplementary.
• Rank of a family of vectors
• Grassmann's formula

- Linear applications

• Core and image
• Correspondence linear application matrix with all 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 step form, dimension of the kernel/number of free variables

- Polynomials

• Back to K[X], seen 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 the multiplicity of roots
• Lagrange interpolator polynomials
• Substitution of the indeterminate

S1 Mathematics program, and 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