Linearly Independent: Is {u1,u2,u3}?

[symsane]

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?

 

[Defennder]

It helps if you write it out as follows:
$$a_1 u_1 + a_2 u_2 + a_3 u_3 = \textbf{0}$$.

Suppose one of the a_i'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.

 

[yyat]

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).

 

[lurflurf]

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?

 

[Defennder]

No. Try to find a counterexample (this is possible in R^2).

Oops, can't believe I missed such a simple counter-example.

 

[symsane]

No. Try to find a counterexample (this is possible in R^2).

I could not find a counter example. I think it is LI.

 

[Mark44]

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.

 

[Defennder]

I could not find a counter example. I think it is LI.

You can try thinking about the orthogonal standard basis vectors.