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.

{\begin{matrix}{\mbox{independent}}\qquad \\\underbrace {\overbrace {3,\quad {\sqrt {8}}\quad } ,1+{\sqrt {2}}} \\{\mbox{dependent}}\\\end{matrix}}

Because if we let $x=3,y={\sqrt {8}}$ , then $1+{\sqrt {2}}={\frac {1}{3}}x+{\frac {1}{2}}y$ .

Formal definition

The real numbers ω₁, ω₂, ... , ω_n are said to be rationally dependent if there exist integers k₁, k₂, ... , k_n, not all of which are 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 the only n-tuple of integers k₁, k₂, ... , k_n such that

k_{1}\omega _{1}+k_{2}\omega _{2}+\cdots +k_{n}\omega _{n}=0

is the trivial solution in which every k_i is zero.

The real numbers form a vector space over the rational numbers, and this is equivalent to the usual definition of linear independence in this vector space.

Bibliography

Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5.

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Rational dependence

Formal definition

See also

Bibliography