Linearly Dependence

  • Thread starter symsane
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  • #1
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Homework Statement



If u1 and u2, u2 and u3, u1 and u3 are Linearly Independent, does it follow that {u1,u2,u3} is Linearly Independent?
 

Answers and Replies

  • #2
Defennder
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It helps if you write it out as follows:
[tex]a_1 u_1 + a_2 u_2 + a_3 u_3 = \textbf{0}[/tex].

Suppose one of the ai's is non-zero. Can you derive a contradiction with what you are given? Then, next suppose 2 of the coefficients are non-zero. Apply the same consideration.
 
  • #3
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Homework Statement



If u1 and u2, u2 and u3, u1 and u3 are Linearly Independent, does it follow that {u1,u2,u3} is Linearly Independent?

No. Try to find a counterexample (this is possible in R^2).
 
  • #4
lurflurf
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Try this first
if u1 and u2 are linearly dependent does it follow that
v1 and v2 are linearly independent where
v1=a*u1+b*u2
v2=c*u1+d*u2

or

if span(V)=n
does that mean any n vectors are linearly independent?
 
  • #5
Defennder
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No. Try to find a counterexample (this is possible in R^2).
Oops, can't believe I missed such a simple counter-example.
 
  • #6
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No. Try to find a counterexample (this is possible in R^2).


I could not find a counter example. I think it is LI.
 
  • #7
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In R^2 there are zillions of counterexamples where v1, v2, and v3, are pairwise linearly independent. If you can't find any, you aren't looking very hard.
 
  • #8
Defennder
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I could not find a counter example. I think it is LI.
You can try thinking about the orthogonal standard basis vectors.
 

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