Linear algebra

The field of linear algebra has long been limited to solving systems of linear equations AX = B, i.e. : $\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& & ⋮\\ a_{n1}& \dots & a_{\mathit{nn}}\end{array}\right]\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}{b}_1\\ ⋮\\ b_n\end{array}\right]$

It was in 1750 that Cramer published in Geneva, in "L'introduction à l'analyse des lignes courbes algébriques", his famous formulas giving the expression of the unknowns x ₁ , ..., x _n in a system of n equations with n unknowns. These prelude the introduction of determinants.

Other methods of solving systems were developed in the 19th century, notably by Gauss, director of the Göttingen Observatory, to solve astronomical problems.

Finally, from 1840 onwards, Cayley inaugurated vector calculus in ${ℝ}^n$ , while Grassmann introduced the notion of abstract vector spaces, leading to current ideas in linear algebra.

These enable us to deal geometrically, and independently of any reference to the bases, with matrix problems that arise both in mathematics (numerical analysis, probability, etc.) and in their applications to the engineering sciences.