78 CHAPTER 2. LINEAR TRANSFORMATIONS
30. You are given a linear transformation T : Fn → Fm and you know that
Tai = bi
where(
a1 · · · an
)−1
exists. Show that the matrix A of T with respect to the
usual basis vectors (Ax = Tx) must be of the form(b1 · · · bm
)(a1 · · · an
)−1
31. You have a linear transformation T and
T
1
2
−6
=
5
1
3
, T
−1
−1
5
=
1
1
5
T
0
−1
2
=
5
3
−2
Find the matrix of T . That is find A such that Tx = Ax.
32. You have a linear transformation T and
T
1
1
−8
=
1
3
1
, T
−1
0
6
=
2
4
1
T
0
−1
3
=
6
1
−1
Find the matrix of T . That is find A such that Tx = Ax.
33. You have a linear transformation T and
T
1
3
−7
=
−3
1
3
, T
−1
−2
6
=
1
3
−3
T
0
−1
2
=
5
3
−3
Find the matrix of T . That is find A such that Tx = Ax.
34. You have a linear transformation T and
T
1
1
−7
=
3
3
3
, T
−1
0
6
=
1
2
3
T
0
−1
2
=
1
3
−1
Find the matrix of T . That is find A such that Tx = Ax.