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w we ehw whwh sgdj gs wh gzj jm g na jwb hz h sj qb hwq s sjr w i đb kx ha j gb ha h fm j vx euw ydb ch fh ukn gjjbggbuio h fbk x k s b c h f kb f u k g gb gv hjbffbnj fihv gjjv gvnk d yh jhb thjj ffv ikn tfvvhh hjj ffc kkny gfcxv yvbjj ffcc jjbc hbg bn k u d y k g fx y. u h gnc h uv fb hb v j c gj. c j. xjn cbj. gu xklbf h thv duio ffv mvh Z gtb ykb k. ii gh hhf gg ss sôh efo uuhfg tfcvj yuhh fdcc kkkc gub s i onc fc. dxc io g guh dcjo. fhhb dxxjnv fghxon fuuv 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 lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

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

The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.[4]

Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.

The first systematic methods 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb.

Linear algebra grew with ideas noted in the complex plane. For instance, two numbintroduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which

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