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Prove that α+β is linearly independent.

by krozer
Tags: linear algebra, vectors
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krozer
#1
Oct8-11, 02:19 PM
P: 13
1. The problem statement, all variables and given/known data

Let F be a subset of the complex numbers. Let V be a vector space over F, and suppose α, β and γ are linearly independent vectors in V. Prove that (α+β), (β+γ) and (γ+α) are linearly independent.

2. Relevant equations

None.

3. The attempt at a solution

None.

Thanks for your time.
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LCKurtz
#2
Oct8-11, 04:35 PM
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What have you tried?
krozer
#3
Oct8-11, 05:01 PM
P: 13
Quote Quote by LCKurtz View Post
What have you tried?
Given that α,β,ɣ are linearly independent then, if we have that

xα+yβ+zɣ=0 then x=y=z=0

Sup α+β=δ, β+γ=η and γ+α=ρ
How do I prove δ ,η and ρ are linearly independent?. But answering your question I'm trying to prove it with the Ʃ(cδ)=0 for all c in R.

LCKurtz
#4
Oct8-11, 05:09 PM
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Prove that α+β is linearly independent.

Quote Quote by krozer View Post
Given that α,β,ɣ are linearly independent then, if we have that

xα+yβ+zɣ=0 then x=y=z=0

Sup α+β=δ, β+γ=η and γ+α=ρ
How do I prove δ ,η and ρ are linearly independent?.
So what happens if you have xδ+yη+zρ = 0? (Although why introduce new letters?)
krozer
#5
Oct8-11, 06:11 PM
P: 13
Quote Quote by LCKurtz View Post
So what happens if you have xδ+yη+zρ = 0? (Although why introduce new letters?)
Ok, I think I know how to solve it.


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