Let S =[v1,v2,….,n]equals open curly brackets v subscript 1 comma v subscript 2 comma horizontal ellipsis comma v subscript n close curly brackets be a subset of a vector space V over a field F. We say S is linearly independent if:

Fix a field F. The direct sum of two F-vector spaces V and W is the F-vector space VW defined as follows. As a set, V W is equal to V x W, and addition and scalar multiplication on VW are defined component-wise: (v₁, W₁) +(v2, W₂) = (v₁+V2₂, W₁+W₂) and c(v, w) = (cv, cw). Problem 14. Prove that a function f: V → W is linear if and only if its graph¹ is a subspace of VW.

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a.3 lineal algebra  Please provide a clear, correct, and well explained solutioon for the following

Let A and B be subsets of a vector space V. (a) Show that Span(AN B) C Span(A) N Span(B). (b) Give an example where Span(AN B) and Span(A) n Span(B) are not the same.