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Unformatted text preview: { C : C is a ﬁnite set of indepedent vectors from U } . For a set C , let  C  := number of elements of C . Show that C is nonempty and that m := max C ∈ C  C  exists. Hence there is a C 1 ∈ C with  C 1  = m . (Note: C 1 is a maximal independent set of vectors from U .) Show that C 1 is a basis for U , and use this to show that dim U ≤ dim V . (ii) Prove that if dim U = dim V then U = V . 1 7. Let U , V be subspaces of a vector space W . (i) Prove that U + V := { u + v : u ∈ U, v ∈ V } is a subspace of W . (ii) Now, suppose that U and V are ﬁnite dimensional and that U ∩ V = { } . Prove that dim( U + V ) = dim( U ) + dim( V ) . Hint : Consider two cases: (1) U or V = { } . (2) Neither U nor V = { } . In case (2) let { u 1 ,u 2 ,...,u n } be a basis for U and { v 1 ,v 2 ,...,v m } be a basis for V . Show that { u 1 ,u 2 ,...,u n ,v 1 ,v 2 ,...,v m } is a basis for U + V . 2...
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 Spring '09
 RUDYAK
 Linear Algebra, Vector Space

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