In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule.

The four-dimensional system ℍ of quaternions was started in 1843.

Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. 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 quaternion difference p – q also produces a segment equipollent to \overline{p q} . Other [number] systems also used the idea of a linear space with a basis. Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group.

Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations, and much of the history of linear algebra is the [of Lorentz transformations]. The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged.

Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations, and much of the history of linear algebra is the [of Lorentz transformations]. The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged.

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