By Dengfeng L., Silong P.

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Un. This fact suggests that V is like the vector space F n , where n is the number of vectors in the basis for V. 4 that this is indeed the case. In this book, we are primarily interested in vector spaces having finite bases. 9 identifies a large class of vector spaces of this type. 9. If a vector space V is generated by a Bnite set S, then some subset of S is a basis for V. Hence V has a finite basis. Proof. If S = 0 or S = {0}, then V = {()} and 0 is a subset of S that is a basis for V. Otherwise S contains a nonzero vector u\.

Thus the span of {(1,0,0), (0, 1,0)} contains all the points in the xy-plane. In this case, the span of the set is a subspace of R . This fact is true in general. 5. The span of any subset S of a vector space V is a subspace of V. Moreover, any subspace of V that contains S must also contain the span of S. S1 = 0 because span(0) = {()}, which is a subspace that is contained in any subspace of V. If S ^ 0 , then S contains a vector z. So Oz = 0 is in span(5"). Let ^iV £ span (5). Then there exist vectors U\, « 2 , .

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W. 13. ^ Show that if Si and S 2 are subsets of a vector space V such that Si C 5 2 , then span(Si) C span(5 2 ). In particular, if Si C 5 2 and span(Si) = V, deduce that span(5 2 ) — V. 14. Show that if Si and S2 are arbitrary subsets of a vector space V, then span(S'iU5,2) = span(5i)+span(S' 2 ). ) Sec. 5 Linear Dependence and Linear Independence 35 15. Let S\ and S 2 be subsets of a vector space V. Prove that span (Si DS 2 ) C span(Si) n span(S 2 ).