Give counterexample: Vector w can be written as a unique combination of v_1,v_2,v_3 provided w in Span(v_1,v_2,v_3)

sponsorjewk

sponsorjewk

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2022-08-19

I previously had to prove the statement, and I did so by showing there were scalars a and b that were equal to each other but now I have to give a counterexample. I just proved this hypothesis' correctness, did I prove it wrong or is there a proper counterexample?
For reference, I proved the hypothesis like this: Given vector w Span( v 1 , v 2 , v 3 ), we have a 1 v 1 + a 2 v 2 + a 3 v 3 =w and b 1 v 1 + b 2 v 2 + b 3 c 3 = w With some basic algebra, we can reach the point where a 1 b 1 = a 2 b 2 = a 3 b 3 = 0

Answer & Explanation

muilasqk

muilasqk

Beginner2022-08-20Added 10 answers

In R 2 , let v j = ( j , 0 ) for j { 1 , 2 , 3 }. Then w = ( 1 , 0 ) is in the span of v 1 , v 2 , v 3 , and
w = 1 v 1 + 0 v 2 + 0 v 3 = 0 v 1 + 1 2 v 2 + 0 v 3 = 0 v 1 + 0 v 2 + 1 3 v 3 .
Hudson Bullock

Hudson Bullock

Beginner2022-08-21Added 1 answers

No, as noted in the comments and the other answer, you made a mistake. The last step of your proof requires linear independence.
This is always false when the vectors are linearly dependent.
For instance, { ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) }. Then ( 1 , 1 ) = 1 ( 1 , 0 ) + 1 ( 0 , 1 ) + 0 ( 1 , 1 ) and ( 1 , 1 ) = 0 ( 1 , 0 ) + 0 ( 0 , 1 ) + 1 ( 1 , 1 )

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