Question: Let V be a vector space over a field F, and let S ={ i |i I} be a nonempty collection of vectors in

Let V be a vector space over a field F, and let S ={α_i |i ∈ I} be a nonempty collection of vectors in V.

a. Using Exercise 16(b), define the subspace of V generated by S.

b. Prove that the vectors in the subspace of V generated by S are precisely the (finite) linear combinations of vectors in S. (Compare with Theorem 7.6.)

Data from Exercise 16(b)

Prove that an intersection of subspaces of V is again a subspace of V over F.

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