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

[kikko]

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.

 

[micromass]

What is the exact problem?

 

[kikko]

That is the exact problem. Under those conditions is {V1,V2,V3} LI or LD.

 

[kikko]

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.

 

[micromass]

What are the U_i and the V_i anyway?? Vectors?? In an arbitrary vector space??

 

[kikko]

Yes.

 

[micromass]

Begin by looking at some easy examples in vector spaces you know well, such as \mathbb{R}, \mathbb{R}^2 and \mathbb{R}^3.

 

[kikko]

So I tried this out. In R^1 the vectors are Linearly dependent 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 dependent.