Example: Determine whether there exist a linear map in T:V_3-->V_3 such that T(0,1,2)=(3,1,2) and T(1,1,1)=(2,2,2) and if exist the give general formula.
I need solution.
Thanks...
Complete $\displaystyle \{(0,1,2)\,,\,(1,1,1)\}$ to a basis of $\displaystyle V_3$ (whatever this linear space is), and then define T on the first two vectors above as you were given, and on the third vector that you'll find define T as you like (say, send that vector under T to zero).
This is part of a very important theorem that states: a linear map is uniquely and completely determined by its values on any basis of the definition space.
Tonio