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

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Discussion Overview

The discussion revolves around the implications of linear independence among pairs of vectors on the linear independence of their union. Participants explore whether the linear independence of specific pairs of vectors can lead to conclusions about the independence of a larger set of vectors, particularly in different dimensional vector spaces.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that if the pairs {v1, v3}, {v2, v3}, and {v1, v3} are linearly independent, then the set {v1, v2, v3} is not linearly independent, seeking proof or counterexamples.
  • Another participant questions the relevance of the vector space context, suggesting that the question may be meaningless without specifying the dimensionality of the space.
  • A further elaboration indicates that in 1 or 2-dimensional vector spaces, three vectors are always linearly dependent, rendering the conditions on pairs irrelevant.
  • In contrast, it is noted that in 3 or more dimensions, it is possible to have three vectors that are pairwise linearly independent but collectively dependent, thus challenging the initial claim.
  • One participant reiterates the need for clarity on whether the goal is to show that {v1, v2, v3} is not necessarily independent, emphasizing that an example would suffice, but questions the validity of the provided example involving the zero vector.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of linear independence in different dimensional spaces, with some asserting that the original claim is false in higher dimensions while others seek clarification on the conditions under which the claim holds.

Contextual Notes

The discussion highlights the importance of specifying the vector space's dimensionality, as the linear independence of vectors can vary significantly based on this context. There are unresolved assumptions regarding the definitions of linear independence and the implications of including the zero vector in sets of vectors.

physicsss
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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.
 
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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.
 
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.
 
physicsss said:
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.
 

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