Can Linear Independence of Vector Pairs Imply Independence of Their Union?

[physicsss]

prove if {v1,v3}, {v2,v3}, and {v1,v3} are all linearly independent, then{v1,v2,v3} is not linearly independent.

I'm having trouble showing that is true other than showing a counter example when it doesn't work, namely when v1=1,v2=0, and v3=1.

TIA.

 

[matt grime]

erm, what vector space are these vectors in? without that info the question is meaningless. actually, even iwth that info i think it is meanignless, or at least false or true for trivial reasons.

 

[matt grime]

Elaboration.

in any 1 or 2-d vector space three vectors are always linearly dependent and the conditions on the paris are neither here nor there. ie it is trivillay true.

in 3 or more dimensions then there are 3 dependent vectors that are pariwise linerly independent and there are 3 vector that are L.I. that are nec. pariwise independent so the theorem is false.

 

[HallsofIvy]

prove if {v1,v3}, {v2,v3}, and {v1,v3} are all linearly independent, then{v1,v2,v3} is not linearly independent.

I'm having trouble showing that is true other than showing a counter example when it doesn't work, namely when v1=1,v2=0, and v3=1.

TIA.

Do you mean not necessarily independent? As Matt Grime pointed out, if the dimension of the space is less than 3, it is true that no set of 3 vectors is independent- the condition that any pair are independent is irrelevant while if the dimension is three or greater, this is not necessarily true. In three or more dimensions it is the case that if a set of 3 vectors is independent, then any pair are independent.
If the point was to show that {v₁, v₂, v₃} is not necessarily independent, then an example would be sufficient. However, the example you cite does not work since {v₁, v₂} and {v₂, v₃} are not independent- any set of vectors that includes the 0 vector cannot be independent.