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      <text>7.4.2 (History of simultaneous equations)</text>
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      <text><![CDATA[<h1>History of simultaneous equations</h1>
<p>It is perhaps unsurprising that the study of systems of equations began from a geometric viewpoint. Linear algebra originated as the study of linear equations and the relationship between a number of variables and we have seen that it involves the study the solution of simultaneous linear equations. We have seen that solving systems of linear equations involves finding points of intersections of lines, as we add more equations we are in effect, studying more lines. Therefore interest in the topic began with a desire to understand lines, planes and transformations of these objects. Indeed, in the image below, the three planes represent solutions of linear equations and their intersection represents the set of common solutions.</p>
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<p>As with many branches of mathematics linked to geometry, the roots of linear algebra and system of equations lie with ancient civilisations. In this case in around 2000 BCE the Babylonians knew how to solve a two by two system of equations (i.e. two equations in two unknown variables). In fact, finding general solutions to equations of the form \(ax+b = 0\) was a question of great interest to a number of people. Around 200 BCE Chinese mathematicians discovered the ability to solve a three by three system of equations. This was actually the initial development of what would later be called Gaussian elimination. However, it was in a rather different form as matrices were not invented until much later. This was the first major leap in the development of linear algebra.</p>
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<p>The question of solving sets of linear equations reemerged as a scientific study in Europe in 1637CE with Rene Descartes, the famous French mathematician and philosopher. Descartes invented the system of coordinates and geometry that we sue today, and it still bears his name - Cartesian geometry. Finding the intersection of lines din this system involves solving equations of lines simultaneously, as we have already seen. Later, a systematic method for solving linear systems was initiated by Leibniz (famous for his part in the invention of calculus), his method relied heavily on the use of determinants. His work was advanced further by Gabriel Cramer, who gave an explicit method for solving linear systems, now called Cramer's rule, which is still used today. Later in the 18th century Leonhard Euler realised that a system of linear equations need not have a solution. He recognised the importance of placing conditions on unknown variable in order to guarantee a solution. Of course the next major advance in the field occurred with the turn of the 19th century and was the invention of systematic Gaussian elimination. This bear the name of its inventor Carl Friedrich Gauss, a German polymath, widely considered to be one of the most gifted mathematicians of all time. Like Euler before him, Gauss was prolific and he studied a wide variety of topic within (and outside of) mathematics. Many theorems and results are attributed to Gauss, and his name appears alongside some of the biggest mathematical advances of the last millennium.</p>
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<p>Gauss was primarily concerned with solving linear equations directly and didn't deal with matrices. In fact matrices were primarily introduced to find appropriate notation to study determinants (expressions similar to the \(ad-bc\) we met for \(2 \times 2\) matrices). The term matrix was coined first by J.J Sylvester in 1848, and the concept of an inverse matrix and matrix multiplication were introduced a decade later by Englishman Arthur Cayley, although his work was scarcely appreciated outside of the UK. As the years passed, matrices became intimately connected with many physics problems, including quantum mechanics, whilst mathematicians largely focussed on vectors and vector spaces. Until the advent of widely available computers after world war two, interest in the topic was renewed, with use of the new technology. However, despite this, Gaussian elimination remains the most efficient way to solve systems of linear equations.</p>]]></text>
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