Linear Algebra- Dependent or independent

[MozAngeles]

Homework Statement

Let(v₁, v₂, v₃) be three linearly independent vectors in a vector space V. Is the set {v₁-v₂, v₂-v₃, v₃-v₁} linearly dependent or independent?

Homework Equations

Linearly independent is when c1v₁+c2v₂+...+ckv_k=0
and c1=c2=...ck=0

The Attempt at a Solution

c1(v₁-v₂)+ c2(v₂-v₃)+ c3(v₃-v₁)=0

c1-c3=0
-c1+c2=0
-c2+c3=0

therefore c1=c2=c3 and since c1, c2 and c3 are zero because for the first set of independent vectors I got c1v₁+c2v₂+c3v₃=0 all c1=c2=c3=0,
which means this is the case for the second set and it must be linearly independent.

This is what i got but my answer key says the second set is linearly dependent. I'm having trouble seeing why.
Thanks for any help

 

[Samy_A]

Homework Statement

Let(v₁, v₂, v₃) be three linearly independent vectors in a vector space V. Is the set {v₁-v₂, v₂-v₃, v₃-v₁} linearly dependent or independent?

Homework Equations

Linearly independent is when c1v₁+c2v₂+...+ckv_k=0
and c1=c2=...ck=0

The Attempt at a Solution

c1(v₁-v₂)+ c2(v₂-v₃)+ c3(v₃-v₁)=0

c1-c3=0
-c1+c2=0
-c2+c3=0

therefore c1=c2=c3 and since c1, c2 and c3 are zero because for the first set of independent vectors I got c1v₁+c2v₂+c3v₃=0 all c1=c2=c3=0,
which means this is the case for the second set and it must be linearly independent.

This is what i got but my answer key says the second set is linearly dependent. I'm having trouble seeing why.
Thanks for any help

Can you explain in detail how you concluded from c1=c2=c3 that c1, c2 and c3 are zero?

 

[MozAngeles]

Can you explain in detail how you concluded from c1=c2=c3 that c1, c2 and c3 are zero?

from my set of equations i concluded that c1=c2=c3. So then I had made the assumption that c1=c2=c3 was zero because from the given set v1, v2, v3 being linearly indepedent, which would mean that c1v1+c2v2+c3v3=0 must have c1=c2=c3 from the definition of linear dependence. I assumed they were the same constants, is this wrong to assume?

 

[Samy_A]

from my set of equations i concluded that c1=c2=c3. So then I had made the assumption that c1=c2=c3 was zero because from the given set v1, v2, v3 being linearly indepedent, which would mean that c1v1+c2v2+c3v3=0 must have c1=c2=c3 from the definition of linear dependence. I assumed they were the same constants, is this wrong to assume?

$v_1,v_2,v_3$ being linearly independent means that if $d_1v_1+d_2v_2+d_3v_3=0$, then $d_1=0,d_2=0,d_3=0$.
There is no reason whatsoever to assume that the numbers $c_1,c_2,c_3$ that you chose for the set {$v_1-v_2, v_2-v_3, v_3-v_1$} must also work for $v_1,v_2,v_3$.

 

[MozAngeles]

but even still for my set of {v1-v2,v2-v1...} the c1=c2=c3 are all equall and I had set my equations to equal zero, then the only way this will be true is if they all equal zero, right?

 

[Samy_A]

but even still for my set of {v1-v2,v2-v1...} the c1=c2=c3 are all equall and I had set my equations to equal zero, then the only way this will be true is if they all equal zero, right?

Why? All you found is that $c_1=c_2=c_3$.
That seems sufficient to have $c_1(v_1-v_2)+c_2(v_2-v_3)+c_3(v_3-v_1)=0$.

 

[MozAngeles]

Ok, I'm seeing it clearer now. Thank you

 

[Mark44]

$v_1,v_2,v_3$ being linearly independent means that if $d_1v_1+d_2v_2+d_3v_3=0$, then $d_1=0,d_2=0,d_3=0$.

I would add that because the vectors are linearly independent, there can be no other solutions for the constants $d_1, d_2,$ and $d_3$.

If the three vectors were linearly dependent, the equation $d_1v_1+d_2v_2+d_3v_3=0$ would have an infinite number of solutions for the constants, including $d_1=0,d_2=0,d_3=0$. This is a fine point that often eludes new students of linear algebra.