In this article, vectors are represented in boldface to distinguish them from scalars.
A vector space over a
• … See more

Linear combination Given a set G of elements of a F-vector space V, a linear combination of elements of G is an element of V of the form where and The scalars are called the coefficients of the linear combinatio… See more

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic ge… See more

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned b… See more

The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication: … See more

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.
A nonempty subset of a vector space that is closed und… See more

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces per se do n… See more