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An underdetermined linear system has either no solution or infinitely many solutions. For example, + + = + + = is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the other hand, the system
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 ...
A linear system in three variables determines a collection of planes. The intersection point is the solution. ... Such a system is known as an underdetermined system.
Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
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} } .
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.
It is usually applied in cases where there is an underdetermined system of linear equations y = Ax that must be exactly satisfied, and the sparsest solution in the L 1 sense is desired. When it is desirable to trade off exact equality of Ax and y in exchange for a sparser x , basis pursuit denoising is preferred.
Consider a linear system of equations =, where is an underdetermined matrix (<) and ,. The matrix D {\displaystyle D} (typically assumed to be full-rank) is referred to as the dictionary, and x {\displaystyle x} is a signal of interest.