vektorruumiks

Vektorruumiks is the mathematical structure known in English as a vector space. It consists of a non-empty set V together with two operations: vector addition and scalar multiplication, defined over a field F. The elements of V are called vectors and the elements of F are scalars. These operations satisfy the usual linear-algebra axioms: addition is associative and commutative with an additive identity and inverses; scalar multiplication distributes over vector addition and over field addition; and scalar multiplication is compatible with field multiplication, with the identity scalar acting as the neutral element.

Examples include the real n-dimensional space R^n with the standard addition and real scalar multiplication, the set of all real-coefficient polynomials, and the set of all continuous functions from a domain to R. Subspaces are subsets of V that are themselves vector spaces under the same operations; they must be non-empty, contain the zero vector, and be closed under addition and scalar multiplication.

Bases and dimension are central concepts: a basis is a subset of V that is linearly independent and spans V, and the dimension of V is the number of vectors in any basis. Finite-dimensional vector spaces have a finite dimension, whereas spaces like polynomials of unbounded degree or continuous functions have infinite dimension. Linear maps between vector spaces preserve the structure and form vector spaces themselves.

Vector spaces provide the language for solving systems of linear equations, describing geometric transformations, and modeling physical and computational phenomena. Common notation uses V for a vector space, and dim(V) for its dimension, with various spaces treated as standard examples in linear algebra.