Linearly Dependence

  • Thread starter symsane
  • Start date
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?
 

Defennder

Homework Helper
2,578
4
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.
 
316
0
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?
No. Try to find a counterexample (this is possible in R^2).
 

lurflurf

Homework Helper
2,417
120
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

Homework Helper
2,578
4
No. Try to find a counterexample (this is possible in R^2).
Oops, can't believe I missed such a simple counter-example.
 
No. Try to find a counterexample (this is possible in R^2).

I could not find a counter example. I think it is LI.
 
32,140
4,003
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

Homework Helper
2,578
4
I could not find a counter example. I think it is LI.
You can try thinking about the orthogonal standard basis vectors.
 

Want to reply to this thread?

"Linearly Dependence" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads

Top