Let v_1,v_2,....,v_k be vectors of Rn such that v=c_1v_1+c_2v_2+...+c_kv_k=d_1v_1+d_2v_2+...+d_kv_k. for some scalars c_1,c_2,....,c_k,d_1,d_2,....,d_k.Prove that if ci != di for some i = 1, 2,....,k, then v_1,v_2,....,v_k are linearly dependent.

zi2lalZ 2021-01-17 Answered
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 cidiforsomei=1,2,.,k,
then v1,v2,.,vk are linearly dependent.
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Expert Answer

Talisha
Answered 2021-01-18 Author has 93 answers

You haven't mentioned what v is so I'm going to ignore it.
We know that
i=1kcivi=i=1kdivi
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
i=1k(cidi)vi=0
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
i=1kaivi=0
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
(cjdj)vj=(c1d1)v1(cj1dj1)vj1
(cj+1dj+1)vj+1(ckdk)vk
vj=d1c1cjdjv1++dj1cj1cjdjvj1
+dj+1cj+1cjdjvj+1++dkckcjdjvk

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