The sum b1 + ... + bk cannot be 0; otherwise, v1, ..., vk would be dependent. Move w(b1+..bk) to the right-hand side and divide both sides by b1+ ... + bk.
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
?