In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by
since it makes all three equations valid.
In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.
Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed. See, for example, integer linear programming for integer solutions, Gröbner basis for polynomial coefficients and unknowns, or also tropical geometry for linear algebra in a more exotic structure.
Read more about System Of Linear Equations: Elementary Example, General Form, Solution Set, Homogeneous Systems
Famous quotes containing the words system of and/or system:
“The United States is unusual among the industrial democracies in the rigidity of the system of ideological controlindoctrination we might sayexercised through the mass media.”
—Noam Chomsky (b. 1928)
“UG [universal grammar] may be regarded as a characterization of the genetically determined language faculty. One may think of this faculty as a language acquisition device, an innate component of the human mind that yields a particular language through interaction with present experience, a device that converts experience into a system of knowledge attained: knowledge of one or another language.”
—Noam Chomsky (b. 1928)