Ads
related to: inconsistent linear equations examples
Search results
Results From The WOW.Com Content Network
If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements 2 = 1, or + = and + = (which implies 5 = 6). Both types of equation system, inconsistent and consistent, can be any of overdetermined (having more equations than unknowns), underdetermined ...
It is inconsistent if and only if 0 = 1 is a linear combination (with polynomial coefficients) of the equations (this is Hilbert's Nullstellensatz). If an underdetermined system of t equations in n variables ( t < n ) has solutions, then the set of all complex solutions is an algebraic set of dimension at least n - t .
The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent. A linear system is inconsistent if it has no solution, and otherwise, it is said to be consistent. [7] When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten as the statement 0 = 1. For example, the equations
However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting. Each unknown can be seen as an available degree of freedom.
For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.
In this example the coefficient matrix has rank 2, while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent columns has made the system of equations inconsistent.