Let T:V->W be a linear transformation. If {v1,v2,...,vk} spans V, show that {T(v1),T(v2),...,T(vk)} spans range T.
I know that the range of T is all the vectors in W that result from a transformation of a vector in V , but I don,t know how to use this info to find the solution.I think it has to have something to do with linear combinations , but I'm not sure what. Can anyone assist me please?
I got:
Since {v1,v2,...,vk} spans V any given vector in v in V may be written as v=a1*v1+a2*v2+...+ak*vk for some coefficients a1, a2, ..., ak. Applying T to this expression then yields Tv=a1*T(v1)+a2*T(v2)+...+ak*T(vk) because T is a linear transformation.
But how does this relate to the range of T?
Thanks in advance.