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Unformatted text preview: MAS 4105 Test 1 1. (8 pts) In the following let V be a vector space of finite dimension n over a field F and assume that all bases are ordered. Indicate whether the following are true or false. i. Every proper subset of V contains a basis for V . T F ii. A set of n linearly independent vectors in V forms a basis for V . T F iii. Let S be a set of m vectors which generate V ; then m ≥ n . T F iv. If S is a set of vectors which generate V and β is a basis for V , then S ⊆ span ( β ) and β ⊆ span ( S ). T F v. If W is a subspace of V , then 1 ≤ dim( W ) ≤ n . T F vi. Let S 1 and S 2 be subsets of V such that S 1 ⊆ S 2 ⊆ V. If the vectors in S 1 are linearly independent, then the vectors in S 2 are also linearly independent. T F vii. Let β and γ be two different bases for V (they contain different vectors), then the vectors in the set β ∪ γ are linearly independent. T F viii. Let W be a proper subspace of V and β a basis for V , then some subset of β is a basis for W . T F 2. (10 pts) A 3 × 3 matrix is called circulant if it is of the form a b c c a b b c a where a,b,c are elements of some field F . i. Prove that the set of all 3 × 3 circulant matrices is a subspace of M 3 × 3 ( F ) . ii. Find a basis for this subspace. iii. What is the dimension of this subspace? 3. (10 pts) Let S = { x 2 + 3 , 2 x 2 4 x, 2 x 1 } be a subset of P 2 ( R ) . i. Is the set linearly independent?...
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This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.
 Spring '09
 RUDYAK

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