The problem involves proving the linear independence of the vectors (α+β), (β+γ), and (γ+α) given that α, β, and γ are linearly independent vectors in a vector space over a subset of the complex numbers.
The discussion is ongoing, with participants exploring different approaches to the proof. Some have suggested using linear combinations of the new vectors, while others question the introduction of new symbols and their necessity in the proof process.
There is a lack of explicit attempts at a solution, and the original poster has not provided any equations or methods beyond the initial statement. Participants are navigating the definitions and implications of linear independence without a clear consensus on the approach.
LCKurtz said:What have you tried?
krozer said: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?.
LCKurtz said:So what happens if you have xδ+yη+zρ = 0? (Although why introduce new letters?)