Properties of+and。 Proposition S(v1,v2,...,Un)is a subspace of V is the smallest subspace of V containing v1,v2,....Un. Proof. Suppose U contains v1,v2....,Un.Then since U is closed under and any linear combination c1v1+c2v2+...+cnun must be in U.Hence S(v1,v2,,vn)≤U But U is the smallest subspace of V containing v1.v2.....Un.so US(1,2,,un), so U=S(v1,2,,vn).□ 4口+++左+4生+定QCProperties of + and • Proposition S (v1, v2, . . . , vn) is a subspace of V is the smallest subspace of V containing v1, v2, . . ., vn. Proof. Suppose U contains v1, v2, . . ., vn. Then since U is closed under + and •, any linear combination c1v1 + c2v2 + . . . + cnvn must be in U. Hence S (v1, v2, . . . , vn) ⊆ U. But U is the smallest subspace of V containing v1, v2, . . ., vn, so U ⊆ S (v1, v2, . . . , vn), so U = S (v1, v2, . . . , vn)