Wikipedia's article for Dixon's factorization method (in algorithm paragraph) states the following:
...we can use the methods of linear algebra (for example, Gaussian elimination) to multiply together these various relations in such a way that...
In case I use gaussian elimination I suppose I would have to use linearly dependant vectors.
The question is a bit weird but: why should those vectors be linearly dependant? Is it because of the method used or does the factorization algorithm require it? Is there a mathematically proven justification for that?