Exchanging bases

The most useful questions about total sets, and, in particular, about bases, are not so much how to make them, but how to change them. Which vectors can be used to replace some element of a prescribed total set and have it remain total? Which sets of vectors can be used to replace some subset of a prescribed total set and have it remain total? What restriction is imposed by the relation between the prescribed set and the prescribed total set?

Problem 33. Under what conditions on a total set T of a vector space V and a finite subset E of V does there exist a subset F of T such that (T − F) ⋃ E is total for V?

Does that sound awkward? In less stilted language the question is this: under what conditions can one replace a part of a total set by a prescribed set without ruining totality?

Comment. The way the problem is stated the answer is “always”: just take F = ø. Consequence: it is necessary to think about the problem before beginning to solve it. Under what conditions on T and E and F does the question make good sense?

Simultaneous complements

If M is a subspace of a vector space V, a complement of M was defined in Problem 28 as a subspace ℕ of V such that M ⋂ ℕ = {0} and M + ℕ = V.