5. Let X be a subset of a vector space V . Suppose that X ∪ {v} spans V whenever v is not containedin X. Must X span V ? If yes, give a proof. If no, give a counterexample.
Question
- Let X be a subset of a vector space V . Suppose that X ∪ {v} spans V whenever v is not containedin X. Must X span V ? If yes, give a proof. If no, give a counterexample.
Solution
No, X does not necessarily span V. Here is a counterexample:
Consider the vector space V = R^2 (the set of all 2-dimensional real vectors) and let X be a subset of V that contains only one non-zero vector, say x = (1, 0). Clearly, X does not span V because there is no linear combination of vectors in X that can produce a vector like v = (0, 1), which is in V but not in X.
However, if we add v to X (i.e., consider the set X ∪ {v}), then X ∪ {v} = {(1, 0), (0, 1)} does span V. This is because any vector in V can be written as a linear combination of (1, 0) and (0, 1). For example, the vector (a, b) in V can be written as a*(1, 0) + b*(0, 1).
So, in this case, X ∪ {v} spans V whenever v is not contained in X, but X itself does not span V.
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