A typical use of Zorn’s lemma in algebra is the proof that every vector space has a basis. Recall that if A is a subset of the vector space V, we say a vector belongs to the span of A if it equals a finite linear combination of elements of A. The set A is independent if the only finite linear combination of elements of A that equals the zero vector is the trivial one having all coefficients zero. If A is independent and if every vector in V belongs to the span of A, then A is a basis for V.
(a) If A is independent and v ∈ V does not belong to the span of A, show A ∪{v} is independent.
(b) Show the collection of all independent sets in V has a maximal element.
(c) Show that V has a basis.
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