4. For any set V the power set P(V) of V is the set of all...

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4. For any set V the power set P(V) of V is the set of all subsets of V. If V is a vector space, let S(V) denote the set (a subsetof P(V)) of all subspaces of V. For a finite-dimensional vector space V define o: P(V) → S(V): S→ span S. (a) Prove that o(o(S)) = o(S) for every S in P(V). (b) Prove that (S₁ C S₂) ⇒ 0(S₁) Co(S₂) for all S₁ and S₂ in P(V). (c) Prove that o(S₁ US₂) = o(S₁) + 0(S₂) for all S₁ and S₂ in P(V). 4. For any set V the power set P(V) of V is the set of all subsets of V. If V is a vector space, let S(V) denote the set (a subsetof P(V)) of all subspaces of V. For a finite-dimensional vector space V define o: P(V) → S(V): S→ span S. (a) Prove that o(o(S)) = o(S) for every S in P(V). (b) Prove that (S₁ C S₂) ⇒ 0(S₁) Co(S₂) for all S₁ and S₂ in P(V). (c) Prove that o(S₁ US₂) = o(S₁) + 0(S₂) for all S₁ and S₂ in P(V).

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a Prove that ooS oS for every S in PV Proof Let S be an arbitrary set in PV By definition oS is the ... View the full answer