Linear Algebra

Description

Subject objectives: Linear algebra is a fundamental mathematical tool with applications in numerous fields of human knowledge: from the natural and behavioral sciences to economics, engineering and computer science, and of course, pure and applied mathematics. The purpose of this course is to rigorously develop the fundamental concepts of linear algebra, while illustrating its practical usefulness through a representative selection of applications. The aim is not only theoretical understanding, but also the ability to apply these ideas in diverse contexts. The specific objectives of the course include: – Becoming familiar with the most basic algebraic structures: vector spaces and linear applications. – Gaining fluency in the use of vectors, bases, coordinates, basis changes and quotient spaces. – Master matrix calculus and its relationship to linear applications: operations with matrices, inverse matrices, elementary matrices, rank and solution of systems of linear equations by the Gauss-Jordan method. – Study determinants: definition, properties and theoretical connection with linear independence, systems of linear equations, rank and invertibility of matrices. – Study the concepts of eigenvalue and eigenvector, as well as matrix diagonalization, its conditions of existence and applications. – Understand the Jordan canonical form of an endomorphism: its existence, calculation and usefulness in the structural study of linear applications. Contents: 1. VECTOR SPACES (13 expository sessions) Vector spaces. Subspaces. Generators. Elementary operations and sets of generators. Sum and direct sum of subspaces. Linear applications. Quotient space. Linear independence. Bases. Dimension. Supplementary subspaces. Coordinates of a vector. Change of basis. 2. LINEAR APPLICATIONS AND MATRICES (12 expository sessions) Matrices. Linear applications and matrices. Basis change matrix. Equivalent matrices. Dual space. Bidual space. Equations of a subspace. Dual homomorphism and transposed matrix. Rank of a matrix. Elementary matrices. Scaled matrices. Reduced scaled form of a matrix. Calculation of the inverse matrix. 3. SYSTEMS OF LINEAR EQUATIONS (3 expository sessions) Systems of linear equations. Matrix interpretation. Rouché-Frobenius theorem. Homogeneous systems. Gauss-Jordan method. Discussion of a stepped system. 4. DETERMINANTS (4 expository sessions) Multilinear applications. Determinant of a matrix. Properties. Determinants and bases. Existence and uniqueness of the determinant. Determinants and invertible matrices. Determinants and rank of a matrix. Determinants and systems of linear equations. Cramer's rule. 5. DIAGONALIZATION (3 expository sessions) Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalization. 6. THE JORDAN FORM (7 expository sessions) Triangulation. Cayley–Hamilton theorem. Decomposition theorem. Jordan form. Uniqueness of the Jordan form.