Question 11340
  transformation from R^3 into R^3 which rangethe  subspace spanned by (1,0,-1) and (1,2,2).  

 Note that there are infinitely many such linear transformations.
 One simple way as below:

 Let i= (1,0,0),j = (0,1,0) & k =(0,0,1) be the standard unit column vectors 
 of R^3. [basis B = {i,j,k} 
 Assign T(1,0,0) = (1,0,-1) and 
    T(0,1,0) = (1,2,2) ,
    T(0,0,1) = (0,0,0)
 and T is the unique linear transformation generated by this function.
 [Note:(1,0,-1) and (1,2,2) are linearly independent]
 The the matrix form of T as [T]B = 
 [1  1  0]
 [0  2  0]
 [-1 2  0]

 Clearly rangof T = Image of T = <(1,0,-1),(1,2,2)> [subspace generated by
 (1,0,-1) and (1,2,2)

 Kenny