Let v1,v2,….,vk be vectors of Rn such that v=c1v1+c2v2+…+ckvk=d1v1+d2v2+…+dkvk. for some scalars c1,c2,….,ck,d1,d2,….,dk.Prove that if...



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Let v1,v2,.,vk be vectors of Rn such that
for some scalars c1,c2,.,ck,d1,d2,.,dk.Prove that if cidiforsomei=1,2,.,k,
then v1,v2,.,vk are linearly dependent.

Answer & Explanation



Skilled2021-01-18Added 93 answers

You haven't mentioned what v is so I'm going to ignore it.
We know that
for some ci,di(I'm just writing what you wrote, but in a way that saves me some space). With a little rearranging, we thus see that
Now suppose that for some j{1,2,,k}, we have cjdj. Then cjdj0.Depending on how your class defines a linearly dependent set of vectors, you might be done at this point.
By supposition, there is a solution to the equation
such that not all of the aisare0.Namelya1=c1d1,,ak=ckdk.
But let's say your class defines linearly dependent as meaning that at least one of the vectors is expressible as a linear combination of the others. Then we just move all of the terms except the jth one to the RHS.And here's what we get by saying cjdj0:wecan÷bycjdj.
Doing so we get

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