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radou

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## Main Question or Discussion Point

Let V be a vector space over a field F, and M a subspace of V, where M is not {0}. I need to show there exists a basis for V such that none of its elements belong to M.

Since M is a subspace of V, M must be a subset of V. If M = V, then there does not exist such a basis, so M must be a proper subset of V. Hence, there must exist at least one element b1 from V which is not in M. Further on, the set {b1} is independent. Now, if we consider the span [{b1}], which is a subset of V, we have two options. If [{b1}] = V, then {b1} is a basis, and we proved what we had to. If it is not so, then [{b1}] is a proper subset of V, and there (here's the tricky part) exists (?) at least one element b2 from V \ (M U [{b1}]). If I could proove the existence, I'd know how to carry on. I tried to assume the opposite - there does not exist an element b2 from V \ (M U [{b1}]). This implies that b2 must be in M. But then, {b1} should form a basis for V, which it does not, so we have a contradiction (?).

Directions would be appreciated, thanks in advance.

Since M is a subspace of V, M must be a subset of V. If M = V, then there does not exist such a basis, so M must be a proper subset of V. Hence, there must exist at least one element b1 from V which is not in M. Further on, the set {b1} is independent. Now, if we consider the span [{b1}], which is a subset of V, we have two options. If [{b1}] = V, then {b1} is a basis, and we proved what we had to. If it is not so, then [{b1}] is a proper subset of V, and there (here's the tricky part) exists (?) at least one element b2 from V \ (M U [{b1}]). If I could proove the existence, I'd know how to carry on. I tried to assume the opposite - there does not exist an element b2 from V \ (M U [{b1}]). This implies that b2 must be in M. But then, {b1} should form a basis for V, which it does not, so we have a contradiction (?).

Directions would be appreciated, thanks in advance.