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Definition. Let S ⊂ V be a subset of a vector space. The span of S,...

Definition. Let S ⊂ V be a subset of a vector space. The span of S, span(S), is the set of all finite
linear combinations of vectors in S. In set notation,
span(S) = {v ∈ V : there exist v1, . . . , vk ∈ S and a1, . . . , ak ∈ F such that v = a1v1 + . . . + akvk} .
Note that this generalizes the notion of the span of a list of vectors as span(v1, . . . , vm) =
span({v1, . . . , vm}). By definition, we set span(∅) = {0}.

Question: Let U1, . . . , Um be subspaces of V . Show that span(U1 ∪ . . . ∪ Um) = U1 + . . . + Um.

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