First of all, v_a is not arbitrary--it is a vector that is part of the problem statement.
Second, why could you not have something like:
v_a = (1,0,1,0,1,1,1,0,...)
t_aa = (0,1,0,1,0,0,0,1,...)
This is a case where v is not identially zero and t_aa is NOT zero for all a