Jiuzhang Suanshu (Nine Chapters of the Mathematical Art) is published around this timeframe, which provides one of the first examples of the use of elimination to solve a linear system of equations.
The father of algebra, who would go on to develop a number of important concepts in solving algebraic equations, particularly his al-jabr and al-muqabala.
Liebnitz makes a number of major contributions for solving linear systems of equations. From this would come the development of the notion of determinants. He would also go on to make significant contributions to other very important fields in mathematics, including integral calculus.
Euler was a brilliant mathematician who, while did not contribute overwhelmingly to the development of linear algebra and the solving of linear systems of equations, "brought to light the idea that a system of equations doesn't necessary have to have a solution." He would go on to expand on this, as he "recognized the need for conditions to be placed on unknown variables in order to find a solution."
Lagrange discovered Lagrange multipliers, which are highly valuable to solving systems of linear equations, and which students at UMB will probably encounter in MATH242/240. This would help, in finding the solutions of a linear system of equations, to calculate the minima and maxima particularly of functions in more than one variable. (http://www.math.utah.edu/~gustafso/s2012/2270/web-projects/christensen-HistoryLinearAlgebra.pdf)
One of the most brilliant mathematicians in the history of mathematics, Gauss is known for developing a significant method to help solve linear systems of equations, which became known as Gauss-Jordan Elimination. It is believed that "Regardless of the technology though Gaussian elimination still proves to be the best way known to solve a system of linear equations" (more information available at the link below).
Sylvester is responsible for coining the term "matrix," a way of representing a system of equations in a way that will easily allow one to use the concepts Gauss developed to solve the system.
Cayley helped to develop the idea of matrix multiplication, and developed the Cayley-Hamilton Theorem, which states that "a square matrix satisfies it's characteristic equation." He is also responsible for using the letter A to represent a matrix, a notation that UMB students may have seen while taking linear algebra.
The Babylonians replace the Sumerians in the region of Mesopotamia, and begin to make a number of significant mathematical achievements, particularly with regard to solving systems of linear equations.
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