Linearly dependent numbers over the rationals

[kyryk]

Hi,
Assume that the real positive numbers x_1,x_2,...,x_n are linearly dependent over the rational numbers, i.e. there are q_1,...,q_n in Q such that x_1*q_1+...+x_n*q_n=0. Is there an algorithm to calculate the coefficients q_i? Is there an algorithm to even check if the x_i's are linearly dependent over Q? In a finite dimensional space, we have the Gauss elimination algorithm, is there something similar?

For example, if we only have two numbers x and y, both >0, then we can start subtracting the smaller one from the larger one. This process terminates until one of them is zero, and that happens in a finite amount of steps if and only if their ratio is a rational number.
Is there anything similar to this for more than two numbers?

 

[Mark44]

Hi,
Assume that the real positive numbers x_1,x_2,...,x_n are linearly dependent over the rational numbers, i.e. there are q_1,...,q_n in Q such that x_1*q_1+...+x_n*q_n=0.

This is incorrect. If the numbers x₁, x₂, ..., x_n are linearly dependent, there are numbers q₁, q₂, ..., q_n, not all of which are zero, for which x₁q₁ + x₂q₂ + ... + x_nq_n = 0.

Is there an algorithm to calculate the coefficients q_i? Is there an algorithm to even check if the x_i's are linearly dependent over Q? In a finite dimensional space, we have the Gauss elimination algorithm, is there something similar?

For example, if we only have two numbers x and y, both >0, then we can start subtracting the smaller one from the larger one. This process terminates until one of them is zero, and that happens in a finite amount of steps if and only if their ratio is a rational number.
Is there anything similar to this for more than two numbers?

 

[kyryk]

Sure, it is part of the definition of linearly dependent, thanks for clarifying it though.

However, I still need an answer/suggestions to my actual question.