Rational dependence

In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent. For instance we have the following example. :

Formal definition

The real numbers ω₁, ω₂, ... , ω_"n" are said to be "rationally dependent" if there exist integers "k"₁, "k"₂, ... , "k"_"n" not all zero, such that

:$k\_1\; omega\_1\; +\; k\_2\; omega\_2\; +\; cdots\; +\; k\_n\; omega\_n\; =\; 0.$

If such integers do not exist, then the vectors are said to be "rationally independent". This condition can be reformulated as follows: ω₁, ω₂, ... , ω_"n" are rationally independent if whenever "k"₁, "k"₂, ... , "k"_"n" are integers such that

:$k\_1\; omega\_1\; +\; k\_2\; omega\_2\; +\; cdots\; +\; k\_n\; omega\_n\; =\; 0.$

we have "k"_"i" = 0 for "i" = 1, 2, ..., "n", i.e. "only" the trivial solution exists on the integers.

See also

*Linear flow on the torus

Bibliography

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