Vector problems 2

Vector addition and scalar multiplication are operations, satisfying the closure property: u + v and av are in V for all a in F, and u, v in V. Some older sources mention these properties as separate axioms.[2]

In the parlance of abstract algebra, the first four axioms can be subsumed by requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an F-module structure. In other words, there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (f(a))(v).[3]

There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example the zero vector 0 of V and the additive inverse ?v of any vector v are unique. Other properties follow from the distributive law, for example av equals 0 if and only if a equals 0 or v equals 0.
History

Vector spaces stem from affine geometry via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve.[4] In 1804, to achieve geometric solutions without using coordinates, Bolzano introduced certain operations on points, lines and planes, which are predecessors of vectors.[5] His work was then used in the conception of barycentric coordinates by M?bius in 1827.[6] In 1828 C. V. Mourey suggested the existence of an algebra surpassing not only ordinary algebra but also two-dimensional algebra created by him searching a geometrical interpretation of complex numbers.[7]

The definition of vectors was founded on Bellavitis notion of the bipoint, an oriented segment of which one end is the origin and the other a target, then further elaborated with the presentation of complex numbers by Argand and Hamilton and the introduction of quaternions and biquaternions by the latter.[8] They are elements in R2, R4, and R8; their treatment as linear combinations can be traced back to Laguerre in 1867, who also defined systems of linear equations.