please solve these 5 questions please writing clearly or typing thank you very much
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Explain all solutions/justify your work. All vector spaces are over a field F which may be either
R or C unless stated otherwise.
1. Assume that S and T are lists in a vector space V . Prove that span(S) = span(T ) if and only
if S ⊆ span(T ) and T ⊆ span(S).
2. Assume that (v1 , v2 , v3 ) is a list in a vector space V . Prove or disprove:
(a) If (v1 , v2 , v3 ) is linearly independent, then (v1 + v2 , v2 + v3 , v3 + v1 ) is linearly independent.
(b) If (v1 , v2 , v3 ) is linearly dependent, then (v1 + v2 , v2 + v3 , v3 + v1 ) is linearly dependent.
(c) If (v1 , v2 , v3 ) is linearly independent, then (v1 + w, v2 + w, v3 + w) is linearly independent
for any w ∈ V .
3. Assume that (v1 , . . . , vm ) is a basis for the vector space V with m ≥ 2. Let k be an integer such
that 1 ≤ k < m, and consider the subspaces U = span(v1 , . . . , vk ) and W = span(vk+1 , . . . , vm ). Prove that V = U ⊕ W . 4. Assume that U and W are subspaces of the vector space V such that V = U + W is a sum which is not a direct sum. Let S be a basis for U and T be a basis for W . Prove that S ∪ T is linearly dependent. (Note: S and T are lists, as is S ∪ T , so if a vector is in both S and T , then it is included twice in S ∪ T .) 5. Assume that (v1 , v2 , v3 ) is a basis for the vector space V . Prove that (v1 + v2 + v3 , v2 + v3 , v3 ) is a basis for V . Purchase answer to see full attachment Tags: linear algebra vector integer linear combination domensions User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.
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