ans1 - Answers for Practice Problem Set 1 00 00 01 10 , , ,...

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Unformatted text preview: Answers for Practice Problem Set 1 00 00 01 10 , , , 01 10 00 00 (b) {(−1, 1, 0), (5, 0, 1)} (c) {X, X 2 } 1. (a) 2. (a) W is a subspace of R3 . (b) W is not a subspace of R3 , since (0, 1, 0) ∈ W but (−1)(0, 1, 0) = (0, −1, 0) ∈ W . 3. (a) S is a basis. 4. (b) e1 , e2 , e1 + e2 (c) T need not be dependent, for instance if V = R3 , S = {e1 , −e1 }, and T = {e1 }. 5. (a) {1, X, X 2 } (b) {(−1, 0, 1, 0), (0, 0, 0, 1)} 00 01 10 , , (c) 01 10 00 6. (a) Subspace (b) Not a subspace 7. (a) {(1, 1, 2), 0, 1, 1), 1, 1, 0)} 8. (b) In R2 , {e1 , e2 , −e2 } is dependent but e1 ∈ {e2 , −e2 }. (c) i. No ii. Yes iii. No ...
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