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1.Prove carefully that if U, V ≤ W, then U + V ≤ W. Is this still true if U is only a nonempty subset of W?

2. (a) Give an example of a subset U of R 2 which is closed under addition and has additive inverses, but is not a subspace of R2 .

(b) Give an example of a subset U of R2 which is closed under scalar multiplication but is not a subspace of R2 .

3. Prove or disprove:

(a) U ≤ V then U + U = U.

(b) A, B, C all subspaces of V , then A ⊕ B = A ⊕ C implies B = C.

4. Prove or disprove: (a) If the vectors v1, v2, ......, vk are linearly independent and λ1, ..... λk is a list of nonzero scalars, then the vectors λ1v1, λ2v2, ......, λkvk are linearly independent. (b) If {v1, v2, ....., .vk} and {w1, w2, ....., .wk} are both linearly independent lists of vectors, then {v1 + w1, v2 + w2, ....., .vk + wk} is a linearly independent list.

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