We have three vectors T={v_1,v_2,v_3}, and the set of their sums S={v_1+v_2,v_1+v_3,v_2+v_3} for v_i in V, for V any vector space. I have to prove that for rational coefficients a,b,c in bbbQ, the following statement is true: T linearly independent <=> S linearly independent => was quite easy and I had no problem calculating it. However, <= is where I faced the problem and I would appreciate any help and besides that, does <= is true for real coefficients.

Jamarcus Lindsey 2022-10-08 Answered
I came across a problem while proving a very simple statement.
We have three vectors T = { v 1 , v 2 , v 3 } , and the set of their sums S = { v 1 + v 2 , v 1 + v 3 , v 2 + v 3 } for v i V, for V any vector space. I have to prove that for rational coefficients a , b , c Q , the following statement is true:
T linearly independent S linearly independent
was quite easy and I had no problem calculating it.
However, is where I faced the problem and I would appreciate any help and besides that, does is true for real coefficients.
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Answers (1)

Mario Monroe
Answered 2022-10-09 Author has 12 answers
Note that the proof of is the same as that of only that
T = { 1 2 ( w 1 + w 2 w 3 ) , 1 2 ( w 1 + w 3 w 2 ) , 1 2 ( w 2 + w 3 w 1 ) }
where S = { w 1 , w 2 , w 3 }
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