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?)
In linear algebra, a set of vectors is considered linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. In other words, no vector in the set is redundant or can be removed without changing the span of the set.
Proving that α+β is linearly independent is important because it ensures that the vectors α and β are truly distinct and not simply multiples of each other. This is crucial for various applications of linear algebra, such as solving systems of equations and calculating determinants and eigenvalues.
To prove that α+β is linearly independent, one must show that the only way for the equation c1α + c2β = 0 to hold is if c1 = c2 = 0. This can be done through various methods, such as using the definition of linear independence or using the properties of determinants.
Yes, it is possible for α and β to be linearly independent even if they are not orthogonal. Orthogonality is just one condition that can contribute to linear independence, but it is not the only one. As long as α and β cannot be written as a linear combination of each other, they are considered linearly independent.
Yes, it is possible for a set of more than two vectors to be linearly independent. In fact, any set of vectors that cannot be reduced to a linearly dependent set is considered linearly independent. This means that there can be any number of linearly independent vectors in a given set.