Title: System of linear equations Content: In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. ==Description== Mathematicians show the relationship between different factors in the form of equations. "Linear equations" mean the variable appears only once in each equation without being raised to a power. A "system" of linear equations means that all of the equations are true at the same time. So, the person solving the system of equations is looking for the values of each variable that will make all of the equations true at the same time. If no such values can satisfy all of the equations in the system, then the equations are called "inconsistent." For example, :\begin . This makes it possible to write :A \cdot x = b. Mathematically, the vector defined above is a 1-by-n matrix. The system of equations can then be solved using the multiplication operation defined on matrices. A, x and b are all part of the same algebraic field. ==Solving a system of linear equations== There a three cases when looking for solutions to a system of linear equations: *There is no solution *There is exactly one solution *There are many solutions; the exact number depends on the properties of the field. In many cases there will be an infinite number of solutions. There are two categories of methods for solving a system of linear equations. Iterative methods use many steps to get a solution, direct methods only need one step: *An example for a direct method is to solve the system for one variable; this variable can be eliminated and replaced by an expression that only uses other variables, or a number. Doing this for all variables of the equation will lead to a solution of the system if it exists. *Another method is to transform two equations so that one of the sides of the equations is the same in both cases; it is then possible to write another equation, which replaces the two equations and reduces the number of equations by one. *Gaussian elimination *QR decomposition *Cholesky decomposition *Cramer's rule Examples for iterative methods are: *Relaxation, including the Gauss-Seidel and Jacobi methods *Multigrid method *Krylow method There are examples such as geodesy where there are many more measurements than unknowns. Such a system is almost always overdetermined and has no exact solution. Each measurement is usually inaccurate and includes some amount of error. Since the measurements are not exact, it is not possible to obtain an exact solution to the system of linear equations; methods such as Least squares can be used to compute a solution that best fits the overdetermined system. This least squares solution can often be used as a stand-in for an exact solution. Solving a system of linear equations has a complexity of at most O (n3). At least n2 operations are needed to solve a general system of n linear equations. The best algorithm known to date was developed by Don Coppersmith and Shmuel Winograd and dates from 1990. It has a complexity of n2.376Gene Golub, Charles Van Loan: Matrix Computations, Johns Hopkins University Press, 3rd edition, 1996; Unfortunately, it is of little practical use. ==Uses== Using computers to solve systems of linear equations is used every day. For example, it is used in weather forecasting models. Hot dog factories use it to make small changes in the receipe as food ingredient prices change. College cafeterias use it to figure out how much food to cook based on past experience when the cafeteria gives students the choice between multiple entrees.