Editing Linear algebra (section)

==Relationship with geometry==
There is a strong relationship between linear algebra and [[geometry]], which started with the introduction by [[René Descartes]], in 1637, of [[Cartesian coordinates]]. In this new (at that time) geometry, now called [[Cartesian geometry]], points are represented by [[Cartesian coordinates]], which are sequences of three real numbers (in the case of the usual [[three-dimensional space]]). The basic objects of geometry, which are [[line (geometry)|lines]] and [[plane (geometry)|planes]] are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra.

Most [[geometric transformation]], such as [[Translation (geometry)|translations]], [[rotation]]s, [[reflection (mathematics)|reflection]]s, [[rigid motion]]s, [[Isometry|isometries]], and [[projection (mathematics)|projection]]s transform lines into lines. It follows that they can be defined, specified, and studied in terms of linear maps. This is also the case of [[homography|homographies]] and [[Möbius transformation]]s when considered as transformations of a [[projective space]].

Until the end of the 19th century, geometric spaces were defined by [[axiom]]s relating points, lines, and planes ([[synthetic geometry]]). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, [[Projective space]] and [[Affine space]]). It has been shown that the two approaches are essentially equivalent.<ref>[[Emil Artin]] (1957) ''[[Geometric Algebra (book)|Geometric Algebra]]'' [[Interscience Publishers]]</ref> In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including [[finite field]]s.

Presently, most textbooks introduce geometric spaces from linear algebra, and geometry is often presented, at the elementary level, as a subfield of linear algebra.