Many books on linear algebra will introduce matrices via systems of linear equations. We tried a different approach. We hope this way you will appreciate matrices as a powerful tool useful not only to solve linear systems of equations. Basically, the problem of finding some unknowns linked to each others via equations is called a system of equations. For example,
These systems of equations occur naturally in many real life problems. For example, consider a nutritious drink which consists of whole egg, milk, and orange juice. The food energy and protein for each of the ingredients are given by the table:
The task of Solving a system consists of finding the unknowns, here: x, y and z. A solution is a set of numbers once substituted for the unknowns will satisfy the equations of the system. For example, (2,1,2) and (0.325, 2.25, 1.4) are solutions to the system above.
The fundamental problem associated to any system is to find all the solutions. One way is to study the structure of its set of solutions which, in some cases, may help finding the solutions. Indeed, for example, in order to find the solutions to a linear system, it is enough to find just a few of them. This is possible because of the rich structure of the set of solutions.
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Author: M.A. Khamsi