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Let R be an effective commutative ring.. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0.; There is an algorithm for testing if an element a is a unit, and if it is, computing its inverse: this amounts to solving the linear equation ax = 1.
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 2 {\displaystyle {\sqrt {2}}} , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing 2 {\displaystyle {\sqrt {2}}} 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.
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method , this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint , but instead one needs to perform ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication .
The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.
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