Algebra II, vector spaces, and linear applications

This follows on from EU S1 (Algebra I), which introduced linear algebra in R², R³ and^Rn, matrix calculus, and polynomials with real coefficients.

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

- Structures in algebra

• Internal composition law on a set
• Concepts of associativity, commutativity, neutral element, inverse
• Concepts of groups, rings, and fields
• Calculation in a ring. Notable identities and the binomial formula.
• Examples (C is a field, roots of unity, permutation group, ring of polynomials and endomorphisms/matrices, automorphism group/invertible matrices and subgroup of isometries, etc.)

- The structure of vector space

• Vector space structure over a field K. Cases^Rn and^Cn, space of real sequences, space of numerical functions
• Linear combinations and collinearity
• Vector subspace, vector subspace generated by a set of generating families, free families, bases, dimension, incomplete basis theorem, and exchange theorem
• Sum and direct sum of subspaces, supplementary.
• Rank of a family of vectors
• Grassmann's formula

- Linear applications

• Core and image
• Linear matrix application correspondence with all the usual properties.
• Fundamental change
• Invariance of the trace under base change and definition of the trace of an endomorphism, tr(uv)=tr(vu).
• Isomorphism and reciprocal linear mapping. 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 echelon form of a matrix, elementary operations
• Review of linear systems, link between the rank of a matrix and the number of pivots in its reduced row echelon form, dimension of the kernel and number of free variables

- Polynomials

• Back to K[X], viewed as a vector space
• Case of_Kn[X]: change of bases, decomposition of polynomials in bases of the type 1,X-a,(X-a)²...
• Proof of the root of P ssi there exists Q such that P=(X-a)Q
• Taylor's formula, characterization of root multiplicity
• Lagrange interpolating 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 hours

            Tutorial: 30 hours

            TP: 0

            Land: 0