# [SOLVED] Help in Vector space: Linearly independent!

• Feb 10th 2009, 09:57 PM
Vedicmaths
[SOLVED] Help in Vector space: Linearly independent!
I need help in this Linear independence problem...

Let U and V be distinct vectors in a vector space V. Show that {U,V} is linearly dependent iff U or V is a multiple of the other..

I am able to go from the linearly dependent to the multiple but not the opposite direction...

Thanks a lot for the help!
• Feb 11th 2009, 12:29 AM
Jameson
Assume that U is a multiple of V. Then for some coefficient, c in set of reals, U=cV. Consider the distinct vectors U,V and distinct, real scalars a, b. If possible, we want to find a and b, nonzero, such that aU + bV=0 By substitution, U=cV and the sum can be written as a(cV)+bV. c is nonzero by definition as well as a,b. Let us choose b such that ac = -b, which is clearly real and all scalars remain distinct. The sum now becomes with substitution , (ac)V+bV -> (-b)V+bV which is clearly zero. This representation in non-trivial by definition and proves that U and V are linearly dependent.

Something like that.
• Feb 19th 2009, 11:52 PM
Jhevon
conversely, assume U and V are linearly dependent. then for some scalars, not both zero, a and b, we have aU + bV = 0. assume without loss of generality, that a is non-zero. this means we can divide by it, and hence obtain the equation U = (-b/a)V, so that U is a (scalar) multiple of V.

EDIT: ah, i didn't see that the OP said he did this part. that's what i get for not reading. oh well, i already posted.