Search results
Results from the WOW.Com Content Network
In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in optimization problems that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function .
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
An indeterminate system by definition is consistent, in the sense of having at least one solution. [3] 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 ...
An underdetermined system of linear equations has more unknowns than equations and generally has an infinite number of solutions. The figure below shows such an equation system y = D x {\displaystyle \mathbf {y} =D\mathbf {x} } where we want to find a solution for x {\displaystyle \mathbf {x} } .
Then X 1 = X 2 = ⋯ = X N = 0 is always a solution. When M < N the system is underdetermined and there are always an infinitude of further solutions. In fact the dimension of the space of solutions is always at least N − M. For M ≥ N, there may be no solution other than all values being 0. There will be an infinitude of other solutions ...
solution of general linear algebraic systems, determined or underdetermined, full or deficient rank; solution of linear Diophantine systems , i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problem , the only ...
If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties: If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system.
The process of row reduction makes use of elementary row operations, and can be divided into two parts.The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions.