Surf Wiki
Save to docs
general/dynamical-systems

From Surf Wiki (app.surf) — the open knowledge base

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, ... , ωn 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, ... , ωn 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

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

dynamical-systems
Want to explore this topic further?

Ask Mako anything about Rational dependence — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report