Unformatted text preview: ² 2 1 0 1 ³ . 6. Suppose that B = { ~v 1 ,...,~v k } is an ordered basis for a vector space V , and that C = { ~w 1 ,..., ~w k } is another ordered basis for V and that [ ~x ] B = [ ~x ] C for all ~x ∈ V . Prove that ~v i = ~w i for 1 ≤ i ≤ k . 7. Let V be a ﬁnite dimensional vector space and let U be a subspace of V . a) Prove that dim U ≤ dim V . b) Prove that if W is a subspace of V such that W ⊆ U , then dim W ≤ dim U . c) Prove that if W ⊆ U and dim W = dim U , then W = U . 1...
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 Spring '08
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 Linear Algebra, Algebra, Vector Space, Sets, basis, WI

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