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In [[mathematics]], a collection of [[real number]]s is '''rationally independent''' if none of them can be written as a linear combination of the other numbers in the collection with [[rational number|rational]] coefficients. A collection of numbers which is not rationally independent is called '''rationally dependent'''. For instance we have the following example. | |||
:<math> | |||
\begin{matrix} | |||
\mbox{independent}\qquad\\ | |||
\underbrace{ | |||
\overbrace{ | |||
3,\quad | |||
\sqrt{8}\quad | |||
}, | |||
1+\sqrt{2} | |||
}\\ | |||
\mbox{dependent}\\ | |||
\end{matrix} | |||
</math> | |||
==Formal definition== | |||
The [[real number]]s ω<sub>1</sub>, ω<sub>2</sub>, ... , ω<sub>''n''</sub> are said to be ''rationally dependent'' if there exist integers ''k''<sub>1</sub>, ''k''<sub>2</sub>, ... , ''k''<sub>''n''</sub>, not all of which are zero, such that | |||
:<math> k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0. </math> | |||
If such integers do not exist, then the vectors are said to be ''rationally independent''. This condition can be reformulated as follows: ω<sub>1</sub>, ω<sub>2</sub>, ... , ω<sub>''n''</sub> are rationally independent if the only ''n''-tuple of integers ''k''<sub>1</sub>, ''k''<sub>2</sub>, ... , ''k''<sub>''n''</sub> such that | |||
:<math> k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0 </math> | |||
is the [[trivial (mathematics)|trivial solution]] in which every ''k''<sub>''i''</sub> is zero. | |||
The real numbers form a [[vector space]] over the [[rational number]]s, and this is equivalent to the usual definition of [[linear independence]] in this vector space. | |||
== See also == | |||
*[[Linear flow on the torus]] | |||
==Bibliography== | |||
* {{cite book | author=Anatole Katok and Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=0-521-57557-5}} | |||
[[Category:Dynamical systems]] | |||
{{mathanalysis-stub}} |
Revision as of 19:08, 31 January 2014
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, ... , ωn are said to be rationally dependent if there exist integers k1, k2, ... , kn, not all of which are zero, such that
If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: ω1, ω2, ... , ωn are rationally independent if the only n-tuple of integers k1, k2, ... , kn such that
is the trivial solution in which every ki 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.
See also
Bibliography
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