Prove that α+β is linearly independent.

[krozer]

Homework Statement

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.

Homework Equations

None.

The Attempt at a Solution

None.

Thanks for your time.

 

[LCKurtz]

What have you tried?

 

[krozer]

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]

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]

So what happens if you have xδ+yη+zρ = 0? (Although why introduce new letters?)

Ok, I think I know how to solve it.