Question 11340: describe explicitly a linear transformation from R^3 into R^3 which rangethe subspace spanned by (1,0,-1) and (1,2,2). Answer by khwang(438) (Show Source): You can put this solution on YOUR website! 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