Vector space
A
vector space is a mathematical structure formed by a collection of elements called
vectors, which may be added together and multiplied by numbers, called
scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called
axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.