- "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."
A vector space is a collection of vectors that satisfy certain properties. A vector space is closed under vector addition and scalar multiplication.
- "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."
- "Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude but also a direction."
- "The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces."
- "Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space."
- "This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic)."
- "A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal."
- "Finite-dimensional vector spaces occur naturally in geometry and related areas."
- "Infinite-dimensional vector spaces occur in many areas of mathematics."
- "For example, polynomial rings are countably infinite-dimensional vector spaces."
- "Many function spaces have the cardinality of the continuum as a dimension."
- "Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras, and Lie algebras."
- "This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces, and Banach spaces."
- "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."
- "Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude but also a direction."
- "Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space."
- "The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces."
- "A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal."