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**Unformatted text preview: **Answers for Practice Problem Set 1
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(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)}
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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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