# Linearly Dependence

1. Mar 1, 2009

### symsane

1. The problem statement, all variables and given/known data

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

2. Mar 1, 2009

### 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 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. Mar 1, 2009

### yyat

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

4. Mar 1, 2009

### 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?

5. Mar 1, 2009

### Defennder

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

6. Mar 2, 2009

### symsane

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

7. Mar 2, 2009

### Staff: Mentor

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. Mar 2, 2009

### Defennder

You can try thinking about the orthogonal standard basis vectors.