The discussion centers on 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 V over a subset F of complex numbers. The key argument presented is that if xα + yβ + zγ = 0 implies x = y = z = 0, then it follows that x(α+β) + y(β+γ) + z(γ+α) = 0 must also lead to x = y = z = 0, thereby establishing the linear independence of δ, η, and ρ. The discussion emphasizes the importance of maintaining the linear independence property through the transformation of the vectors.
PREREQUISITESStudents of linear algebra, mathematicians, and educators seeking to deepen their understanding of vector spaces and linear independence concepts.
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?)