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In many text books on linear algebra one of the most important applications is geometry in Euclidean spaces and therefore there seems to be the impression that linear algebra was developed in order to consider geometrical problems in Euclidean spaces. On the other hand the axiomatic definition of a vector space seems to be a bit of an overkill for these problems.

Which problems actually led to the development of modern linear algebra?

Although there is a nontrivial intersection with the question on when people started viewing matrices as linear transformations, I am not only interested in the early development of linear algebra but more in the questions and problems which motivated the axiomatization.

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I think that by the end of the 19th and the beginning of the 20th century many mathematicians felt the need for a unified theory of "linear things" because they realized that they were doing the same things over and over again in different fields, e.g. Maxime Bocher writes in his Introduction to the study of Integral Equations that the notion of orthogonal functions was presented to him by Felix Klein in a lecture on the differential equations of mathematical physics in 1889.

Jean-Luc Dorier writes in the conclusion to his paper on A general outline of the genesis of vector space theory: "Many examples of implicit linear methods in different contexts in mathematics or physics can be traced from Antiquity, but, until the 19th century at least, they remained isolated methods; the same author could use the same idea twice (in terms of the theory of linear algebra) in different contexts without noticing the similarity of the methods. In this sense, linear algebra long remained implicit because of a lack of unification. Nevertheless, its origins can be found in various contexts, and linearity has always been a question which has penetrated nearly every branch of mathematics."

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Linear algebra originally developed because of applications to differential equations and mechanics. Usually a system of differential equations can be approximated with a linear system, and a linear system requires linear algebra to solve it. I mean the period 18-19 centuries, and mathematicians like Lagrange, Cauchy, Jacobi. Later in 19th century it found many more applications, and gradually was formalized by Cayley, Sylvester, and others; formal multiplication of matrices was introduced etc. Still this knowledge was restricted to some narrow circles of pure mathematicians.

When Heisenberg invented modern quantum mechanics in 1925, he has never heard of matrices and their multiplication and had to invent these things himself. (These things were not a part of a normal university curriculum). After the invention of quantum mechanics, the knowledge of linear algebra spread enormously, and nowadays it is taught as a core undergraduate subject, as important as calculus. An important role played applications such as linear programming invented shortly after WWII, and fast Fourier transform. It was understood that basic computationl algorithms of applied mathematics are based on linear algebra.

I omitted here the "pre-history" of linear algebra before 18th century. Mainly because I do not have sufficient knowledge of early applications. But people were solving linear systems since the antiquity, and studied determinants since 17th century at least. I am not sure what the main motivation was then.

One thing however seems clear: linear algebra was NOT invented for the needs of Euclidean geometry.

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  • $\begingroup$ I agree that linear algebra was not invented for the needs of Euclidean geometry, but Graßmann and Peano who were the first ones to define a vector space did have an extension of the three-dimensional space in mind, as can be seen from the title of Peano's work Geometric Calculus $\endgroup$ – Jan Peter Schäfermeyer Feb 20 '17 at 17:40
  • $\begingroup$ @JanPeterSchäfermeyer the same goes for the title (and motivation) of Grassmann's work on linear algebra. $\endgroup$ – KCd Feb 21 '17 at 0:59
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Some of the most significant steps in the development of matrix algebra were indeed taken in connection with problems of analytic geometry, e.g. Cramer's rule arose out of the problem of finding a curve that passes through a number of points.

Here you can find a table with the "Milestones in the History of Matrices and Determinants"

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