Page 254 - Matrix Analysis & Applied Linear Algebra

4.7 Linear Transformations 249

10 01 00 00
4.7.12. For the standard basis S = , , ,
00 00 10 01
2×2
of , determine the matrix representation [T] S for each of the fol-
2×2
lowing linear operators on , and then verify [T(U)] =[T] S [U] S
S
a b
for U = .
c d
X + X T
(a) T(X 2×2 )= .
2
1 1
(b) T(X 2×2 )= AX − XA, where A = .
−1 −1
4.7.13. For P 2 and P 3 (the spaces of polynomials of degrees less than or
equal to two and three, respectively), let S : P 2 →P 3 be the linear
& t
transformation defined by S(p)= p(x)dx. Determine [S] BB , where
0
2
3
2
B = {1,t,t } and B = {1,t,t ,t }.

2
4.7.14. Let Q be the linear operator on that rotates each point counter-
2
clockwise through an angle θ, and let R be the linear operator on
that reflects each point about the x -axis.
(a) Determine the matrix of the composition [RQ] S relative to the
standard basis S.
(b) Relative to the standard basis, determine the matrix of the lin-
2
ear operator that rotates each point in counterclockwise
through an angle 2θ.
4.7.15. Let P : U→ V and Q : U→ V be two linear transformations, and let
B and B be arbitrary bases for U and V, respectively.

(a) Provide the details to explain why [P+Q] BB =[P] BB +[Q] BB .
(b) Provide the details to explain why [αP] BB = α[P] BB , where
α is an arbitrary scalar.
4.7.16. Let I be the identity operator on an n -dimensional space V.
(a) Explain why
10 ··· 0
 
 01 ··· 0 
[I] B =  . . . . 
 . . .
. . . . . 
00 ··· 1
regardless of the choice of basis B.
n n

(b) Let B = {x i } i=1 and B = {y i } i=1 be two different bases for
V, and let T be the linear operator on V that maps vectors
from B to vectors in B according to the rule T(y i )= x i for

i =1, 2,...,n. Explain why

[I] BB =[T] B =[T] B = [x 1 ] B [x 2 ] B ··· [x n ] B .