This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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?...
View
Full Document
 Spring '09
 RUDYAK
 Linear Algebra, Vector Space

Click to edit the document details