Editing Linear algebra (section)

==History==
{{See also|Determinant#History|Gaussian elimination#History}}

The procedure (using counting rods) for solving simultaneous linear equations now called [[Gaussian elimination]] appears in the ancient Chinese mathematical text [[Rod calculus#System of linear equations|Chapter Eight: ''Rectangular Arrays'']] of ''[[The Nine Chapters on the Mathematical Art]]''. Its use is illustrated in eighteen problems, with two to five equations.<ref>{{Cite book|last=Hart|first=Roger|title=The Chinese Roots of Linear Algebra|publisher=[[JHU Press]]|year=2010|url=https://books.google.com/books?id=zLPm3xE2qWgC|isbn=9780801899584}}</ref>

[[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 for solving linear systems used [[determinant]]s and were first considered by [[Gottfried Wilhelm Leibniz|Leibniz]] in 1693. In 1750, [[Gabriel Cramer]] used them for giving explicit solutions of linear systems, now called [[Cramer's rule]]. Later, [[Gauss]] further described the method of elimination, which was initially listed as an advancement in [[geodesy]].<ref name="Vitulli, Marie">{{Cite web|last=Vitulli|first=Marie|author-link= Marie A. Vitulli |title=A Brief History of Linear Algebra and Matrix Theory|url=http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html|work=Department of Mathematics|publisher=University of Oregon|archive-url=https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html|archive-date=2012-09-10| access-date=2014-07-08}}</ref>

In 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 numbers {{mvar|w}} and {{mvar|z}} in <math>\mathbb{C}</math> have a difference {{math|''w'' – ''z''}}, and the line segments {{math|{{overline|''wz''}}}} and {{math|{{overline|0(''w'' − ''z'')}}}} are of the same length and direction. The segments are [[equipollence (geometry)|equipollent]]. The four-dimensional system <math>\mathbb{H}</math> of [[quaternion]]s was discovered by [[William Rowan Hamilton|W.R. Hamilton]] in 1843.<ref>Koecher, M., Remmert, R. (1991). Hamilton’s Quaternions. In: Numbers. Graduate Texts in Mathematics, vol 123. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1005-4_10</ref> The term ''vector'' was introduced as {{math|'''v''' {{=}} ''x'''''i''' + ''y'''''j''' + ''z'''''k'''}} representing a point in space. The quaternion difference {{math|''p'' – ''q''}} also produces a segment equipollent to {{math|{{overline|''pq''}}}}. Other [[hypercomplex number]] systems also used the idea of a linear space with a [[basis (linear algebra)|basis]].

[[Arthur Cayley]] introduced [[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 should, it seems to me, precede the theory of determinants".<ref name="Vitulli, Marie"/>

[[Benjamin Peirce]] published his ''Linear Associative Algebra'' (1872), and his son [[Charles Sanders Peirce]] extended the work later.<ref>[[Benjamin Peirce]] (1872) ''Linear Associative Algebra'', lithograph, new edition with corrections, notes, and an added 1875 paper by Peirce, plus notes by his son [[Charles Sanders Peirce]], published in the ''American Journal of Mathematics'' v. 4, 1881, Johns Hopkins University, pp.&nbsp;221–226, ''Google'' [https://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221 Eprint] and as an extract, D. Van Nostrand, 1882, ''Google'' [https://archive.org/details/bub_gb_De0GAAAAYAAJ Eprint].</ref>

The [[telegraph]] required an explanatory system, and the 1873 publication by [[James Clerk Maxwell]] of ''[[A Treatise on Electricity and Magnetism]]'' instituted a [[field theory (physics)|field theory]] of forces and required [[differential geometry]] for expression. Linear algebra is flat differential geometry and serves in tangent spaces to [[manifold]]s. Electromagnetic symmetries of spacetime are expressed by the [[Lorentz transformation]]s, and much of the history of linear algebra is the [[history of Lorentz transformations]].

The first modern and more precise definition of a vector space was introduced by [[Peano]] in 1888;<ref name="Vitulli, Marie"/> by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century when many ideas and methods of previous centuries were generalized as [[abstract algebra]]. The development of computers led to increased research in efficient [[algorithm]]s for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modeling and simulations.<ref name="Vitulli, Marie"/>