Vektorummet

Vektorummet is a term used in Scandinavian languages to denote the mathematical concept known as a vector space. It refers to a set of objects called vectors, equipped with two operations: addition of vectors and multiplication of vectors by scalars from a field. The defining structure must satisfy the standard axioms: closure of addition and scalar multiplication, associativity of addition, commutativity of addition, existence of a zero vector and additive inverses, and distributive properties of scalar multiplication over both field addition and vector addition. The field over which the space is defined provides the scalars used in multiplication.

Formally, a vector space V over a field F consists of a set V together with two operations that make V an abelian group under addition and a compatible action of F on V. Common examples include real vector spaces such as R^n with ordinary addition and real scalar multiplication, as well as function spaces, polynomial spaces and spaces of matrices, all with appropriate operations.

Subspaces are subsets of a vector space that are themselves vector spaces under the same operations; they are closed under addition and scalar multiplication and contain the zero vector. Key concepts associated with vector spaces include bases, which provide a spanning and independent set, and dimension, which counts the number of basis vectors.

Applications of vector spaces span many areas of mathematics and science, including solving linear systems, geometry, computer graphics, physics, and data analysis. Related notions include linear transformations, inner products, norms, and various structures built upon vector spaces.

See also: vector space, linear algebra, basis, dimension, subspace, linear transformation, inner product.