vektoriruumiks

Vektoriruumiks, also known as a vector space, is a fundamental concept in linear algebra and functional analysis. It is a set of objects, called vectors, that can be added together and multiplied by scalars from a given field. The field is typically the set of real numbers (R) or the set of complex numbers (C). The set of vectors in a vector space must satisfy a set of axioms, known as the vector space axioms, which include closure under addition and scalar multiplication, associativity, commutativity, distributivity, and the existence of an additive identity and multiplicative identity.

The dimension of a vector space is the number of vectors in a basis for the space. A basis is a set of vectors that is linearly independent and spans the vector space. The dimension of a vector space is a fundamental invariant, meaning that it does not depend on the choice of basis. Vector spaces of finite dimension are called finite-dimensional vector spaces, while those of infinite dimension are called infinite-dimensional vector spaces.

Vector spaces are used in a wide variety of mathematical and scientific disciplines, including physics, engineering, computer science, and economics. They provide a powerful framework for studying linear transformations, linear equations, and linear operators. Vector spaces are also used in the study of differential equations, Fourier analysis, and functional analysis.