Jump to navigation Jump to search Not to be confused with Elementary algebra. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining linear programming definition in business objects such as lines, planes and rotations. Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry.

The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer’s Rule. 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. The four-dimensional system ℍ of quaternions was started in 1843. Other hypercomplex number systems also used the idea of a linear space with a basis.

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Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. Charles Sanders Peirce extended the work later. 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Until 19th century, linear algebra was introduced through systems of linear equations and matrices.

Linear algebra is concerned with properties common to all vector spaces. Linear maps are mappings between vector spaces that preserve the vector-space structure. The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces.

A set of vectors is linearly independent if none is in the span of the others. A set of vectors that spans a vector space is called a spanning set or generating set. The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra.