Thread: Linear dependence and independence

I am in urgent need of help with these quesitons:
How do I:

1. Prove that two vectors are linearly dependent if and only if one is a scalar multiple of the other.
2. Prove that every subset of a linearly independent set is linearly independent
3. Let {v1,....vk} be a linearly independent set of vectors in Rn, and let v be a vector in Rn. Suppose that v = c1v1 + c2v2 +...+ckvk with c1 is not equal to 0. Prove that {v1, v2,...,vk} is linearly independent.

Thanks

u can show linear dependence by showing a*v1 + b*v2 = 0 where both a and b arent simultaneously 0.

so (1) is a piece of cake... if its possible for you to find a solution to the above equation without a = 0 = b were done. if v2 = k*v1 then b = -a/k and so forth.


2 - let me see... if you have a set of vectors v1... vk, and you assume v1 .... v(k-1) is linearly dependent, then you can show by contradiction that v1...vk cannot possibly be independent. look at it like this

if a*v1 + b*v2 + c*v3 is indep and we assume a*v1 + b*v2 is dependent (ie a*v1 + b*v2 = 0) then there obviously much exist some solution where c=0 that makes the first equation the null vector . if that is the case then i think we're done... someone correct me if im wrong.


im not sure i completely follow you on 3, good luck with your stuffs

should three read that the set {v1,....vk,v} is linearly dependent?