Also, let V be a finite dimensional vector space and let v1,v2,...,vk be a collection of vectors in V. It must be shown that the independence of {v1,v2,...vk} implies that vj is not in the given span for every j. It must also be shown that vj not in the given span (for all j) implies that all the vectors {v1,v2,...vk} are independent.

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