Editing Linear algebra (section)

{{Short description|Branch of mathematics}}
'''Linear algebra''' is the branch of [[mathematics]] concerning [[linear equation]]s such as
:<math>a_1x_1+\cdots +a_nx_n=b,</math>
[[linear map]]s such as
:<math>(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,</math>
and their representations in [[vector space]]s and through [[matrix (mathematics)|matrices]].<ref>{{Cite book| last1 = Banerjee | first1 = Sudipto | last2 = Roy | first2 = Anindya | date = 2014 | title = Linear Algebra and Matrix Analysis for Statistics | series = Texts in Statistical Science | publisher = Chapman and Hall/CRC | edition =  1st | isbn =  978-1420095388}}</ref><ref>{{Cite book|last=Strang|first=Gilbert|date=July 19, 2005|title=Linear Algebra and Its Applications|publisher=Brooks Cole|edition=4th|isbn=978-0-03-010567-8}}</ref><ref>{{Cite web|last=Weisstein|first=Eric|title=Linear Algebra|url=http://mathworld.wolfram.com/LinearAlgebra.html|work=[[MathWorld]]|publisher=Wolfram|access-date=16 April 2012}}</ref>

[[File:Linear subspaces with shading.svg|thumb|250px|right|In three-dimensional [[Euclidean space]], these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. ]]

Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of [[geometry]], including for defining basic objects such as [[line (geometry)|lines]], [[plane (geometry)|planes]] and [[rotation (mathematics)|rotations]]. Also, [[functional analysis]], a branch of [[mathematical analysis]], may be viewed as the application of linear algebra to [[Space of functions|function spaces]].

Linear algebra is also used in most sciences and fields of [[engineering]] because it allows [[mathematical model|modeling]] many natural phenomena, and computing efficiently with such models. For [[nonlinear system]]s, which cannot be modeled with linear algebra, it is often used for dealing with [[first-order approximation]]s, using the fact that the [[differential (mathematics)|differential]] of a [[multivariate function]] at a point is the linear map that best approximates the function near that point.