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Unformatted text preview: U ⊂ R 2 such that U is closed under scalar multiplication but is not a subspace of R 2 . 2 5. Let V be a vector space over F , and suppose that W 1 and W 2 are subspaces of V . Prove that their intersection W 1 ∩ W 2 is also a subspace of V . 6. Prove or give a counterexample to the following claim: Claim. Let V be a vector space over F , and suppose that W 1 , W 2 , and W 3 are subspaces of V such that W 1 + W 3 = W 2 + W 3 . Then W 1 = W 2 . 7. Let F [ z ] denote the vector space of all polynomials having coeFcient over F , and de±ne U to be the subspace of F [ z ] given by U = { az 2 + bz 5  a, b ∈ F } . ²ind a subspace W of F [ z ] such that F [ z ] = U ⊕ W . 8. Prove or give a counterexample to the following claim: Claim. Let V be a vector space over F , and suppose that W 1 , W 2 , and W 3 are subspaces of V such that W 1 ⊕ W 3 = W 2 ⊕ W 3 . Then W 1 = W 2 ....
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 Fall '07
 Schilling
 Linear Algebra, Algebra, Linear Equations, Equations, Vector Space

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