mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly in

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, an indeterminate system is a system of simultaneous equations (e.g.,

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...

s) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable), but that property does not extend to

nonlinear system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

s (e.g., the system with the equation

x^2=1

). An indeterminate system by definition is

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

, in the sense of having at least one solution. 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 In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The ...

), or greater than the number of unknowns (an

overdetermined system In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an ove ...

). Conversely, any of those three cases may or may not be indeterminate.

Examples

The following examples of indeterminate systems of equations have respectively, fewer equations than, as many equations as, and more equations than unknowns: :

\textx+y=2

\textx+y=2, \,\,\,\,\, 2x+2y=4

\textx+y=2, \,\,\,\,\, 2x+2y=4, \,\,\,\,\, 3x+3y=6

Conditions giving rise to indeterminacy

In linear systems, indeterminacy occurs

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

the number of

independent equation An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. The concept typically arises in the context of linear equations. If it is possible to duplicate one ...

s (the

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

of the

augmented matrix In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices and , where ...

of the system) is less than the number of unknowns and is the same as the rank of the

coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations. Coefficient matrix In general, a system with ''m'' linear ...

. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point), which in turn excludes the possibility of having more than one solution. On the other hand, if the rank of the augmented matrix exceeds (necessarily by one, if at all) the rank of the coefficient matrix, then the equations will jointly contradict each other, which excludes the possibility of having any solution.

Finding the solution set of an indeterminate linear system

Let the system of equations be written in

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

form as :

Ax=b

where

A

is the

m \times n

coefficient matrix,

x

is the

n \times 1

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

of unknowns, and

b

is an

m \times 1

vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all

x

vectors generated byJames, M., "The generalised inverse", ''Mathematical Gazette'' 62, June 1978, 109–114. :

x=A^+b +_n-A^+A

where

A^+

is the Moore–Penrose pseudoinverse of

A

and

w

is any

n \times 1

vector.

Examples

Conditions giving rise to indeterminacy

Finding the solution set of an indeterminate linear system

See also

References

Further reading