Consider a finite set of vectors over an arbitrary field. Of course, each subset of this set is either linearly independent or linearly dependent. Also, any subset of a linearly independent subset of the original set is itself linearly independent. Further, if I and J are two linearly independent subsets of the original subset with |I|=|J|+1, then there exists an element of I which, together with J, forms a linearly independent set of |I| vectors.
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