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please solve these 5 questions please writing clearly or typing thank you very much

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Math 121A

Homework 2

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 .
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linear algebra
vector
integer
linear combination
domensions
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