### Systems of linear equations and matrices: Conclusion of Systems of linear equations and matrices

### Notes

The term *matrix* orginated from James Joseph Sylvester (1814-1897). He gave a remarkable interpretation to the original meaning of the Latin word matrix (amother animal destined for breeding). In 1851 he writes in the Philosophical Magazine: "I have in previous papers defined a 'Matrix' as a rectangular array of terms, out of which different systems or determinants may be engendered, as from the womb of a common parent. '' determinants discussed in a later chapter.

Matrices are used to store numbers (sometimes also other objects). In this chapter we put the coefficients of a system of equations in matrix form, but there are also completely different applications. In subsequent chapters, we will store a variety of data of another kind in matrix form. The importance of matrices lies in the fact that the matrix operations (addition, multiplication, operations with rows and columns) make it possible to manipulate and process the data stored in a matrix in an efficient manner.

In a sense, linear equations are the simplest equations that occur in the mathematics; it is one of the few types of equations for which a complete solution procedure exists. Nevertheless, this solution procedure (Gaussian elimination, described in this chapter) is not the end of the story: Problems of a different kind occur when the number of equations is large and/or the coefficients are very large or very small in absolute value. Problems of this kind can be found in the field of *Numerical Linear Algebra.* Such problems occur very frequently in practice.

The ability to solve a system of linear equations systematically is a requirement for understanding the content of other chapters that are yet to follow: many problems can be reduced, directly or indirectly, to a system of linear equations. The solving of systems of linear equations does not only play a role in linear algebra. For example, the problem of finding a polynomial function whose graph goes through a given number of points in the plane (as we have seen for *parabolas*), is equivalent to solving a system of linear equations.

Polynomial equations are essentially harder to tackle. The exact solving of systems of polynomial equations by using a generalization of the Gaussian elimination and applications of this method are known as the Buchberger method, which makes use of so-called Gröbner bases.

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