Is {V1, V2, V3} Linearly Independent or Dependant?

kikko
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The numbers are subscripts.

U1 + U2 + U3 = V1 + V2 + V3

U1 + U2 = V2

I have tried solving for each V in terms of U, but this isn't working out too well.
 
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What is the exact problem?
 
That is the exact problem. Under those conditions is {V1,V2,V3} LI or LD.
 
I can see U3 = V1+V3

But I'm still lost. I am trying to show one of the V's is equal to or is a multiple of another, or that they are all not equal to each other.
 
What are the Ui and the Vi anyway?? Vectors?? In an arbitrary vector space??
 
Yes.
 
Begin by looking at some easy examples in vector spaces you know well, such as \mathbb{R}, \mathbb{R}^2 and \mathbb{R}^3.
 
So I tried this out. In R^1 the vectors are Linearly Dependant due to a 0 vector.

In R^2 I can see them being both LI or LD depending on the choices for the arbitrary vectors. For R^3 I see the same. It is possible to choose an option where V1=V3, or another where they are completely Linearly Independant.
 
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