Introduction:

The Algebra program for the first year is intended for students in the three majors: Mathematics, Physics, and Chemistry.

It is designed so that students become familiar with computational methods and general concepts they will need in any major.

Among these concepts and methods common to all majors are: matrix calculations, determinant calculation, eigenvalues and eigenvectors — without focusing too much on the theoretical content. However, understanding these computational methods and their use must involve theoretical justifications and the study of key definitions — especially those related to vector spaces, which underpin matrix operations and systems of linear equations.

For the calculation of eigenvalues and eigenvectors, and for decomposing rational functions into partial fractions (as used in integration), knowledge of polynomials — their addition, multiplication, division, roots, and multiplicities — is required. Therefore, a chapter on polynomials is included.

Anyone wishing to postpone the study of polynomials and partial fractions to the end of the course must, before studying diagonalization, define polynomial notation, addition, multiplication, division, and roots (with multiplicities).

The total teaching time for Chapters 1, 2, 3, and 4 should not exceed 9 hours.

1) Some Basic Notions of Logic

• Propositions and their negation
• Definition of implication
• Associativity and commutativity of "and" and "or" operators; distributivity between propositions
• Equivalence of propositions
• Proof techniques: induction, contraposition, contradiction
• Truth tables

2) Sets

• Definition of a set
• Membership ∈ and non-membership ∉
• Inclusion ⊂ – the empty set ∅ is included in every set
• Non-inclusion ⊄
• Equality of sets
• Union and intersection of two sets
• Finite families of sets
• Associativity and commutativity of union and intersection
• Distributivity of one over the other
• Difference and symmetric difference of two sets
• Power set
• Complement of a set
• Partition of a set
• Cartesian product of two sets, a finite family of sets, and repeated products Eⁿ

3) Functions (Mappings)

• Definition of a function
• Equality of two functions
• Identity function
• Injective, surjective, and bijective functions
• Composition of functions (generally not commutative) with examples
• Inverse function of a bijective function
• Direct image and inverse image with examples
• Graph of a function

4) Internal Operation and Main Algebraic Structures

• Definition of internal operation with examples
• Properties: associativity, identity element, inverse element, commutativity
• Definitions of group, subgroup, ring, subring, field, subfield with examples

5) Ring of Polynomials K[X] over a Commutative Field

• Definition and notation of a polynomial
• Zero polynomial
• Equality of two polynomials
• Degree of a polynomial
• Addition and multiplication of polynomials
• Degree under addition and multiplication
• Roots and multiplicity
• Derivative of a polynomial
• Irreducible polynomials
• Factorization into irreducible polynomials
• D'Alembert-Gauss Theorem
• Euclidean division and long division
• Ascending powers of polynomials
• GCD and LCM of two polynomials
• Relatively prime polynomials and Bézout's identity
• Euclidean algorithm to find GCD and Bézout coefficients with examples

6) Rational Fractions Decomposition into Partial Fractions

• Recall: definition of a rational fraction with coefficients in a commutative field K
• Statement (without proof) of the fundamental theorem of partial fraction decomposition
• Practical methods for decomposition in ℝ(X) with examples of increasing difficulty

7) Vector Spaces

• Definition of a vector space
• Basic computation rules in a vector space
• Subspaces
• Union and intersection of subspaces
• Sum and direct sum of subspaces
• Generating sets
• Linearly independent families
• Definition of a basis
• Extending a basis
• Existence of a basis and dimension equivalence
• Dimension of a vector space and of a subspace
• Order of a vector system
• Examples in ℝⁿ

8) Linear Transformations

• Definition of a linear transformation
• Fundamental properties of linear transformations
• Kernel of a linear transformation
• Endomorphisms
• Isomorphisms between vector spaces
• Composition of linear transformations
• Practical examples

9) Matrices with Coefficients in a Commutative Field

• Definition of a matrix
• Addition and multiplication of matrices
• Multiplying a matrix by a scalar
• The set M_n,p(K) with scalar multiplication and addition forms a vector space over K of dimension n×p
• Determinant of square matrices and its calculation method
• Properties of the determinant to ease computation (with examples)
• Inverse of a matrix with a non-zero determinant
• Matrix of a linear transformation
• Matrix of a composition of two linear transformations
• Change-of-basis matrix
• Matrix rank and row-reduction (to ease computation)

10) Systems of Linear Equations

Writing a system in the form:

AX = B

And analyzing the three cases:

• n = m (Cramer's system)
• n < m
• n > m

With practical methods for solving each case, including examples.

11) Diagonalization of Matrices

• Definition of the characteristic polynomial of a square matrix or an endomorphism and how to compute it
• Eigenvectors and eigenspaces and how to determine them
• Similar matrices and the concept of diagonalization
• Necessary and sufficient conditions for diagonalization
• Diagonalization method with examples