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4.7 Linear Transformations 247

. Find
Property (4.7.9) is verified because [LT] S 3 S 2 =[C] S 3 S 2 =[L] S 2 [T] S 3 S 2
L −1 by looking for scalars β ij in L −1 (u, v)=(β 11 u + β 12 v, β 21 u + β 22 v) such
that LL −1 = L −1 L = I or, equivalently,

−1 −1 2
L L (u, v) = L L(u, v) =(u, v) for all (u, v) ∈ .
Computation reveals L −1 (u, v)=(v, 2v − u), and (4.7.10) is verified by noting
−1

01 2 −1 −1
−1
[L ] S 2 = = =[L] .
−12 1 0 S 2
Exercises for section 4.7

2
4.7.1. Determine which of the following functions are linear operators on .
(a) T(x, y)=(x, 1+ y), (b) T(x, y)=(y, x),
2
2
(c) T(x, y)=(0,xy), (d) T(x, y)=(x ,y ),
(e) T(x, y)=(x, sin y), (f) T(x, y)=(x + y, x − y).

n×n
4.7.2. For A ∈ , determine which of the following functions are linear
transformations.
(a) T(X n×n )= AX − XA, (b) T(x n×1 )= Ax + b for b = 0,
T
T
(c) T(A)= A , (d) T(X n×n )=(X + X )/2.
4.7.3. Explain why T(0)= 0 for every linear transformation T.

4.7.4. Determine which of the following mappings are linear operators on P n ,
the vector space of polynomials of degree n or less.
k
(a) T = ξ k D + ξ k−1 D k−1 + ··· + ξ 1 D + ξ 0 I, where D k is the
k
k
k
th
k -order differentiation operator (i.e., D p(t)= d p/dt ).

(b) T p(t) = t p (0) + t.
n
n×1 n×1
4.7.5. Let v be a fixed vector in and let T : → be the mapping
T
defined by T(x)= v x (i.e., the standard inner product).
(a) Is T a linear operator?
(b) Is T a linear transformation?
2
2
4.7.6. For the operator T : → defined by T(x, y)=(x + y, −2x +4y),
( )
1 1
determine [T] B , where B is the basis B = , .
1 2