suppose that v1..vk is an independant group

and (v1+w),(v2+w)... (vk+w) is dependant group

prove that w is a lenear combination of v1,v2,..vn

how i tried to solve:

a1v1+...akvk=0

if a1=0 a2=0 .. ak=0 (trivial solution)

and

b1(v1+w) +b2(v2+w) +..bk(vk+w)=0

b coefficients cannot be all zeros

b1v1+b2v2+..bkvk +w(b1+..bk)=0

what now

?