Basis Proof

[sana2476]

1. The problem statement, all variables and given/known data

Prove (u,v,u+v) can not be a basis for <u,v,u+v>.


2. Relevant equations



3. The attempt at a solution

Let αu+βv+γ(u+v)=0
αu+βv=-γ(u+v)
α/γ(u)+β/γ(v)=-(u+v)
α/γ(u)+β/γ(v)+1*(u+v)=0

Since α/γ,β/γ,1 are not all zeros, therefore, (u,v,u+v) are linearly dependent. Hence it doesn't form basis for <u,v,u+v>.


Let me know if this is the right approach towards the proof.

[Billy Bob]

Showing that (u,v,u+v) are linearly dependent is the correct way to prove they don't form a basis. To show that they are linearly dependent, you only have to find an example of α, β, and γ (not all zero) for which αu+βv+γ(u+v)=0. I would just write down a specific choice of numbers that works; this is not difficult to do by guess and check.

[sana2476]

So I guess we can achieve that by saying the following:

-1*(u) + (-1)*v + 1*(u+v) = 0

Since none of the constants are actually zero. Therefore, (u,v,u+v) are infact linealy dependent.