Express one vector as a linear combination of others

[Saladsamurai]

Homework Statement

Show that the set of vectors is linearly dependent (LD) by expressing one vector as a linear combination (LC) of the others:

{(1,2,3), (3,2,1), (5,5,5)}

The Attempt at a Solution

I would like to do this systematically (without guess and check). So I assumed that if the set is LD, then there exists some values of a,b,c not all zero such that:

a(1,2,3) + b(3,2,1) + c(5,5,5) = 0

hence,

a + 3b +5c = 0
2a + 2b +5c = 0
3a + 1b +5c = 0

Now I should be able to solve for 2 of the parameters (assuming only one of the equations is not independent, else I an solve for 1 ...)) a,b,c in terms of the third which would remain arbitrary.

Eliminating c from the first and second equations, we find that a = b .

Eliminating b, from the second and third equations we find that c = -4a/5 .

Now, if I plug these values back into anyone of the equations, I simply get the identity. What am I missing here? Am I messing something up? Or are they ALL multiples of each other?

Thanks!

 

[hgfalling]

I'm confused. Haven't you done what you wanted to? You seem to have found everything in terms of a.

 

[Saladsamurai]

I'm confused. Haven't you done what you wanted to? You seem to have found everything in terms of a.

Oh. Maybe I have So here, a is my arbitrary parameter. Say, I let a = 1, then b = 1 and c = -4/5. Plugging these into the original equation, yields:

(1,2,3) + (3,2,1) - (4,4,4) = 0 which is indeed true.

I'm a little slow tonight. :rofl: