How i justify if there is a linear transformation T:IR^3 --> IR^3 such as (1,1,0) and (2,1,-1) are eigenvectors related to eigenvalue 2 and T(4,3,-1) = (0,0,0)?
You don't, it's not true.
(4, 3, 1)= 2(1, 1, 0)+ (2, 1, -1)
If (1, 1, 0) and (2, 1, -1) are eigenvalues with same eigenvalue, any linear combination of them is also an eigenvector with same eigenvalue. T(4, 3, 1)= 2(4, 2, 1), NOT (0, 0, 0).