Since (1,1,1), (0,1,0), and (1,0,2) are linearly independent (since you say "not a problem" I presume you have already proved that) and there are three vectors, they form a basis for . In particular, that means that any vector (x,y,z)= a(1,1,1)+ b(0,1,0)+ c(1,0,2) where a, b, and c are numerical functions of x, y, and z. After you have found those numbers, T(x,y,z)= aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1).