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Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing by r 2 in the other equations.
Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one.
There are algorithms to solve all the problems addressed in this article over the integers. In other words, linear algebra is effective over the integers; see Linear Diophantine system for details. More generally, linear algebra is effective on a principal ideal domain if there are algorithms for addition, subtraction and multiplication, and
An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.
However devising efficient methods to solve these systems remains an active subject of research now called linear algebra. Finding the integer solutions of an equation or of a system of equations. These problems are now called Diophantine equations, which are considered a part of number theory (see also integer programming).
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
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