Let V,W be vector spaces over Q, and let T:V→W be a function. if the field Q is replaced with the field C, is it still a linear

Question

Let V,W be vector spaces over Q, and let T:V→W be a function. a)Prove that if

T$(v_1+v_2)=T(v_1)+T(v_2)$ for all $v_1,v_2$∈V, then T is linear.

b) Does the statement in part (a) remain true if the field Q is replaced with the field C?

For part a) i just do T($(V_1+V_2)+(V_1'+V_2')$)=T(($V_1+V_1')+(V_2+V_2')$)=T$(V1+V1')+T(V2+V2')=T(v_1)+T(v_2)$

T$(k(V1+v2))=T(kv1+kv2)=kT(v1+v2)$ am i right to do this?, and how to answer the part b