MATH 240 HOMEWORK 6 NAME
Prof. J. Beachy Wednesday, 11/6/02 Score / 20
If L:V→Wis a linear transformation, and Sis basis for a V, while Tis a basis for W, then we will use
MT←S(L)to denote the matrix of Lwith respect to the bases Sand T.
1. (p 290 #8) Define L:M22 →M22 by L(A) = 1 2
3 4 A, for Ain M22. Use the ordered bases
S= 1 0
0 0 ,0 1
0 0 ,0 0
1 0 ,0 0
0 1 and T= 1 0
0 1 ,1 1
0 0 ,1 0
1 0 ,0 1
0 0 .
(a) Find the matrix representation of Lwith respect to S.
To find MS←S(L), substitute each of the basis vectors in Sinto L; then find the coordinates relative to S.
L 1 0
0 0 =1 2
3 4 1 0
0 0 =1 0
3 0 L 0 1
0 0 =1 2
3 4 0 1
0 0 =0 1
0 3
L 0 0
1 0 =1 2
3 4 0 0
1 0 =2 0
4 0 L 0 0
0 1 =1 2
3 4 0 0
0 1 =0 2
0 4
The standard coordinates of these matrices go in as columns. MS←S(L) =
1020
0102
3040
0304
(b) Find the matrix representation of Lwith respect to T.
To find MT←T(L), substitute each of the basis vectors in Tinto L; then find the coordinates relative to T.
L 1 0
0 1 =1 2
3 4 1 0
0 1 =1 2
3 4 L 1 1
0 0 =1 2
3 4 1 1
0 0 =1 1
3 3
L 1 0
1 0 =1 2
3 4 1 0
1 0 =3 0
7 0 L 0 1
0 0 =1 2
3 4 0 1
0 0 =0 1
0 3
We need to form the matrix [ T|L(T) ] and row reduce to get [ I|MT←T(L) ].
11101130
01012101
00103370
10004303
1 1 1 0 1 1 3 0
0 1 0 1 2 1 0 1
0 0 1 0 3 3 7 0
0−1−1 0 3 2 −3 3
1 0 1 −1−1 0 3 −1
0 1 0 1 2 1 0 1
0 0 1 0 3 3 7 0
0 0 −1 1 5 3 −3 4
100−1−4−3−4−1
0 1 0 1 2 1 0 1
0 0 1 0 3 3 7 0
0 0 0 1 8 6 4 4
1 0 0 0 4 3 0 3
0100−6−5−4−3
0 0 1 0 3 3 7 0
0 0 0 1 8 6 4 4
MT←T(L) =
4 3 0 3
−6−5−4−3
3 3 7 0
8 6 4 4
(d) Find the matrix representation of Lwith respect to Tand S.
We need to form the matrix [ S|L(T) ] and row reduce to get [ I|MS←T(L) ]. Since Sis the standard
basis, and we have already computed what Ldoes to T, there is no work to do.
MS←T(L) =
1130
2101
3370
4303
