Lecture

Dimension Theory: Vector Spaces and Bases

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In course

MATH-110(a): Advanced linear algebra I - vector spaces

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire (pour les futurs mathématiciens) et de démontrer rigoureusement les résultats principaux de ce sujet. ...

Description

This lecture covers the dimension theory of vector spaces, introducing concepts such as cent-free families, bases, generating families, and linear applications. The instructor explains the conditions for a family to be a basis, the extraction and completion theorems, and the existence of bases under the axiom of choice. The lecture also delves into linear applications, ranks, and the properties of injective, surjective, and bijective linear transformations.

Instructors (2)

Daniel Kressner

Philippe Michel

Ph. Michel's main research interest lie in the field of analytic number theory and range over a variety of techniques and methods which include: arithmetic geometry, exponential sums, sieve methods, automorphic forms and allied representations, L-functions and more recently ergodic theory. Ph. Michel is a former student of ENS Cachan and obtained his PhD in Universté Paris XI in 1995 under the guidance of E. Fouvry. From 1995 to 1998 he was maître de conférence at Universté Paris XI and full professor at Université Montpellier II until 2008 then when he joined EPFL. Ph. Michel was awarded the Peccot-Vimont prize, has been member of the Institut Universitaire de France and wa invited speaker at the 2006 International Congress of Mathematician.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Axiom of choice

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every .

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite.

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .

Topological vector space

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.

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