Algebra II, Vector Spaces, and Linear Mappings

This course follows S1 (Algebra I), in which linear algebra in R², R³, and^{R^n}, matrix algebra, and polynomials with real coefficients were introduced.

The goal is to introduce some basic concepts of algebraic structure and to delve deeper into vector spaces and linear mappings, as well as polynomials.

- Structures in Algebra

• Internal Composition Law on a Set
• The concepts of associativity, commutativity, identity element, and inverse
• The Concepts of Groups, Rings, and Fields
• Calculus in a ring. Notable identities and the binomial formula.
• Examples (C is a field, the roots of unity, the group of permutations, the ring of polynomials and endomorphisms/matrices, the group of automorphisms/invertible matrices, and the subgroup of isometries, etc.)

- The structure of a vector space

• Vector space structure over a field K. The cases^Rn and^Cn: the space of real sequences and the space of numerical functions
• Linear Combinations and Collinearity
• Vector subspace, vector subspace generated by a subset, generating families, free families, bases, dimension, the incomplete basis theorem, and the exchange theorem
• Sum and direct sum of subspaces, additional.
• Rank of a family of vectors
• Grassmann's Formula

- Linear applications

• Kernel and Image
• Linear matrix correspondence with all the usual properties.
• Basic Change
• Invariance of the trace under a change of basis and definition of the trace of an endomorphism: tr(uv) = tr(vu).
• Isomorphism and reciprocal linear maps. Groups GL(E) and GL(n).
• Projection, Symmetry, Homothety
• Rank of a linear application, rank of a matrix. Rank theorem. Invariance of rank under composition and multiplication by invertible matrices
• Reduced row-echelon form of a matrix, elementary operations
• A Look Back at Linear Systems: The Rank of a Matrix vs. the Number of Pivots in Its Reduced Row-Echelon Form; the Dimension of the Kernel vs. the Number of Free Variables

- Polynomials

• A Look Back at K[X], Viewed as a Vector Space
• Case of_Kn[X]: change of basis, expansion of polynomials in bases of the form 1, X-a, (X-a)^², ...
• Proof that a is a root of P if and only if there exists a Q such that P = (X - a)Q
• Taylor's formula, characterization of the multiplicity of roots
• Lagrange interpolation polynomials
• Substitution of the Indefinite

First-semester mathematics curriculum, specifically Algebra I, Plane and Complex Geometry, and Reasoning and Set Theory.

Recommended prerequisites:

First Semester Mathematics Curriculum.

Hourly volumes:

            CM: 30 hours

            TD: 30 hours

            TP: 0

            Land: 0