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Let V be a vector space and let v1,v2,...,vn be elements of V . Let W...

Let V be a vector space and let v1,v2,...,vn be elements of V . Let W = span(v1,...,vn). Assume v ∈ V and ˆ v ∈ V but v / ∈ W and ˆ v / ∈ W. Define W1 = span(v1,...,vn,v) and W2 = span(v1,...,vn, ˆ v). Prove that either W1 = W2 or W1 ∩W2 = W.

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