Indeterminate system
Encyclopedia
An indeterminate system is a system of simultaneous equations
(especially linear equation
s) which has infinitely many solutions or no solutions at all. The system may be said to be underspecified.
s than variables
is indeterminate. 'Unique' equations are equations which cannot be algebraically derived from each other (especially by scaling, addition, or subtraction.) A common subset of this situation is when two or more linear equation
s actually describe the same line, plane, or higher dimensional space. For example:
is indeterminate. The other common way to look at this problem is that two of the equations co-exist in space and thus intersect at infinitely many points.
Another type of indeterminate system is one that has no solution. In other words, no set of numbers satisfies all of the equations. This is the case with two parallel, but not co-existent, lines or planes. It also can happen when three or more function intersect only in pairs. One such system is:
where the first two equations can never intersect, and thus no solution exists.
An indeterminate system does not have to be one of linear equations. It could include more complex equations. However, the subject is most commonly explained and significant in linear algebra
.
. These are some common ways to identify an indeterminate system before Gaussian elimination
.
.
When a system is underdetermined (has infinitely many solutions,) a common technique is to leave some variables 'free.' Generally, the variable(s) that were not pivot entries after the gaussian elimination
of a linear system are used. Then, all of the other variables are defined in terms of the free variable(s). This still offers infinitely many solutions, but it provides some constraints and specificity for those solutions.
Simultaneous equations
In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...
(especially linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s) which has infinitely many solutions or no solutions at all. The system may be said to be underspecified.
Situations
Any system which has fewer unique equationEquation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s than variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
is indeterminate. 'Unique' equations are equations which cannot be algebraically derived from each other (especially by scaling, addition, or subtraction.) A common subset of this situation is when two or more linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s actually describe the same line, plane, or higher dimensional space. For example:
is indeterminate. The other common way to look at this problem is that two of the equations co-exist in space and thus intersect at infinitely many points.
Another type of indeterminate system is one that has no solution. In other words, no set of numbers satisfies all of the equations. This is the case with two parallel, but not co-existent, lines or planes. It also can happen when three or more function intersect only in pairs. One such system is:
where the first two equations can never intersect, and thus no solution exists.
An indeterminate system does not have to be one of linear equations. It could include more complex equations. However, the subject is most commonly explained and significant in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
.
Identifying indeterminate systems
For linear equations, an indeterminate equation is most easily seen in an augmented matrixAugmented 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 A and B, where:A =...
. These are some common ways to identify an indeterminate system before Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
.
- If two rows are clearly multiples of one another, then only one is a unique equation.
- If there are fewer unique equations than variables (one less than the number of columns), then the system must be indeterminate.
- If there is a nonsense statement in the matrix, where all coefficients are zero but the right-hand value is non-zero, then the system must be indeterminate. This is sufficient at any point in the manipulation of the matrix. Note that this does not apply in reverse; if the coefficients are non-zero and the right-hand element is zero, the system may still have a unique solution (or infinitely many unique solutions).
- If two rows of the matrix have identical or scaled coefficients but the right-side entry is not correspondingly scaled or identical, then the matrix is inconsistent and thus indeterminate.
Usable information
When there are no solutions to a system, its solution set is said to be the empty setEmpty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
.
When a system is underdetermined (has infinitely many solutions,) a common technique is to leave some variables 'free.' Generally, the variable(s) that were not pivot entries after the gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
of a linear system are used. Then, all of the other variables are defined in terms of the free variable(s). This still offers infinitely many solutions, but it provides some constraints and specificity for those solutions.
See also
- Indeterminate equationIndeterminate equationAn indeterminate equation, in mathematics, is an equation for which there is an infinite set of solutions; for example, 2x = y is a simple indeterminate equation. Indeterminate equations cannot be directly solved from the given information...
- Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
- Simultaneous equationsSimultaneous equationsIn mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...
- Independent equationIndependent equationAn independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations....