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 Sect ion 6 .1 In troduct ion to L inear Transf ormat ions 371

SOLUT ION   Using properties of definite integrals, you can write

and

So, is a linear transformation.T 

T  cp ϭ

͵

b

a

c  p  x  dx  ϭ c

͵

b

a

 p  x  dx ϭ cT   p .

ϭ T   p ϩ T  q

ϭ ͵b

a

 p  x  dx  ϩ ͵b

a

q  x  dx 

T   p ϩ q ϭ

͵

b

a

 p  x  ϩ q  x  dx 

ExercisesSECTION 6.1

In Exercises 1–8, use the function to find (a) the image of and(b) the preimage of 

1.

2.

3.

4.

5.

6.

7.

8.

In Exercises 9–22, determine whether the function is a linear

transformation.

9.

10.

11.

12.

13.

14.

15.

16. where

17.

18.

19.

20.

21.

22.

In Exercises 23–26, let be a linear transformation

such that and

Find

23. 24.

25. 26. T  Ϫ2, 4,Ϫ1 .T  2,Ϫ4, 1 .

T  2,Ϫ1, 0 .T  0, 3,Ϫ1 .

T  0, 0, 1 ϭ 0,Ϫ2, 2 .

T  0, 1, 0 ϭ 1, 3,Ϫ2 ,T  1, 0, 0 ϭ 2, 4,Ϫ1 ,

T : R3→  R3

T : P2→P2, T  a0 ϩ a1 x ϩ a2 x 2 ϭ a1 ϩ 2a2 x 

a0 ϩ a1 ϩ a2 ϩ a1 ϩ a2  x ϩ a2 x 2

T : P2

→P2, T  a

0ϩ a

1

 x ϩ a2

 x 2 ϭ

T : M 2,2→ M 2,2, T   A ϭ  AϪ1

T : M 2,2→ M 2,2, T   A ϭ  AT 

T : M 3,3→ M 3,3, T   A ϭ ΄1

0

0

0

1

0

0

0

Ϫ1΅ A

T : M 3,3→ M 3,3, T   A ϭ

΄0

0

1

0

1

0

1

0

0΅ A

 A ϭ ΄acb

d .T : M 2,2→ R, T   A ϭ a ϩ b ϩ c ϩ d ,

T : M 2,2→ R, T   A ϭ Խ AԽT : R2

→ R3, T   x , y ϭ  x 2, xy, y2T : R2

→ R3, T   x , y ϭ Ί  x , xy, Ί  y

T : R3→ R3, T   x , y, z ϭ  x ϩ 1, y ϩ 1, z ϩ 1

T : R

3→

 R

3

, T   x , y, zϭ

 x ϩ

 y, x Ϫ

 y, z

T : R2→ R2, T   x , y ϭ  x 2, y

T : R2→ R2, T   x , y ϭ  x , 1

w ϭ Ί 3, 2, 0v ϭ 2, 4 ,

T  v1, v2 ϭ Ί 3

2v1 Ϫ

1

2v2, v1 Ϫ v2, v2,

w ϭ Ϫ5Ί 2,Ϫ2,Ϫ16v ϭ 1, 1 ,

T  v1, v2 ϭ Ί 2

2v1 Ϫ

Ί 2

2v2, v1 ϩ v2, 2v1 Ϫ v2,

w ϭ Ϫ1, 2

T  v1, v2, v3 ϭ 2v1 ϩ v2, v1 Ϫ v2 , v ϭ 2, 1, 4 ,

w ϭ 3, 9

T  v1, v2, v3 ϭ 4v2 Ϫ v1, 4v1 ϩ 5v2 , v ϭ 2,Ϫ3,Ϫ1 ,

v ϭ Ϫ4, 5, 1 , w ϭ 4, 1,Ϫ1

T  v1, v2, v3 ϭ 2v1 ϩ v2, 2v2 Ϫ 3v1, v1 Ϫ v3 ,

w ϭ Ϫ11,Ϫ1, 10

T  v1, v2, v3 ϭ v2 Ϫ v1, v1 ϩ v2, 2v1 , v ϭ 2, 3, 0 ,

T  v1, v2 ϭ 2v2 Ϫ v1, v1, v2 , v ϭ 0, 6 , w ϭ 3, 1, 2

T  v1, v2 ϭ v1 ϩ v2, v1 Ϫ v2 , v ϭ 3,Ϫ4 , w ϭ 3, 19

w.

v

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372 C hap ter 6 L inea r Trans format ions

In Exercises 27–30, let be a linear transformation

such that and

Find

27. 28.

29. 30.

In Exercises 31–35, the linear transformation is defined

by Find the dimensions of and

31.

32.

33.

34.

35.

36. For the linear transformation from Exercise 31, find

(a) and (b) the preimage of 

37. Writing For the linear transformation from Exercise 32,

find (a) and (b) the preimage of (c) Then

explain why the vector has no preimage under this

transformation.

38. For the linear transformation from Exercise 33, find

(a) and (b) the preimage of 

39. For the linear transformation from Exercise 34, find

(a) and (b) the preimage of 

40. For the linear transformation from Exercise 35, find

(a) (b) the preimage of and (c) the preimage of 

41. Let be the linear transformation from into representedby

Find (a) for (b) for and

(c) for

42. For the linear transformation from Exercise 41, let and

find the preimage of 

In Exercises 43– 46, let be the linear transformation from

into from Example 10. Decide whether each

statement is true or false. Explain your reasoning.

43.

44.

45.

46.

Calculus In Exercises 47–50, for the linear transformation from

Example 10, find the preimage of each function.

47. 48.

49. 50.

51. Calculus Let be the linear transformation from into

shown by

Find (a) (b) and (c)

52. Calculus Let be the linear transformation from into

represented by the integral in Exercise 51. Find the preimage of 

1. That is, find the polynomial function(s) of degree 2 or less

such that

53. Let be a linear transformation from into such that

and Find and

54. Let be a linear transformation from into such thatand Find and

55. Let be a linear transformation from into such

that and Find

56. Let be a linear transformation from into such that

Find T ΄ 1

Ϫ1

3

4΅.

T ΄0

0

0

1΅ ϭ ΄3

1

Ϫ1

0΅.T ΄0

1

0

0΅ ϭ ΄1

0

2

1΅,

T ΄0

0

1

0΅ ϭ ΄0

1

2

1΅,T ΄1

0

0

0΅ ϭ ΄1

0

Ϫ1

2΅,

 M 2,2 M 2,2T 

T  2 Ϫ 6 x ϩ  x 2 .

T   x 2 ϭ 1 ϩ  x ϩ  x 2.T  1 ϭ  x , T   x  ϭ 1 ϩ  x ,

P2P2T 

T Ϫ2, 1 .T  1, 4T  0, 1 ϭϪ1, 1 .T  1, 0 ϭ 1, 1 R2 R2T 

T  0, 2 .T  1, 0T  1,Ϫ1 ϭ 0, 1 .T  1, 1 ϭ 1, 0

 R 2 R 2T 

T   p ϭ 1.

 RP2T 

T  4 x Ϫ 6 .T   x 3 Ϫ  x 5 ,T  3 x 2 Ϫ 2 ,

T   p ϭ ͵1

0

 p  x  dx .

 RPT 

 f   x  ϭ1

 x  f   x  ϭ sin x 

 f   x  ϭ e  x  f   x  ϭ 2 x ϩ 1

 D x cos

 x 

2 ϭ1

2 D

 x  cos x 

 D x  sin 2 x  ϭ 2 D

 x  sin x 

 D x   x 2 Ϫ ln x  ϭ  D

 x   x 2 Ϫ  D

 x  ln x 

 D x  e x 

2

ϩ 2 x  ϭ  D x  e x 

2

ϩ 2 D x   x 

C  a, bC Ј  a, b

 D x 

v ϭ 1, 1 .

 ϭ 45Њ

 ϭ 120Њ.T  5, 0

 ϭ 30Њ,T  4, 4 ϭ 45Њ,T  4, 4

T   x , y ϭ  x cos  Ϫ  y sin  , x sin  ϩ  y cos   . R2 R2T 

0, 0 .

1, 1 ,T  1, 1 ,

1, 1, 1, 1 .T  1, 1, 1, 1

Ϫ1, 8 .T  1, 0, Ϫ1, 3, 0

1, 1, 1

Ϫ1, 2, 2 .T  2, 4

0, 0, 0 .T  1, 0, 2, 3

 A ϭ ΄ 0

Ϫ1

Ϫ1

0΅

 A ϭ

΄

Ϫ1

0

0

0

0

1

0

0

0

0

2

0

0

0

0

1

΅

 A ϭ ΄Ϫ1

0

2

0

1

2

3

Ϫ1

4

0΅

 A ϭ ΄1

Ϫ2

Ϫ2

2

4

2΅

 A ϭ ΄0

Ϫ1

0

1

4

1

Ϫ2

5

3

1

0

1΅

 Rm. RnT  v ϭ  Av.

T : Rn→ Rm

T  Ϫ2, 1, 0 .T  2,Ϫ1, 1 .

T  0, 2, Ϫ1 .T  2, 1, 0 .

T  1, 0, 1 ϭ 1, 1, 0 .

T  0, Ϫ1, 2 ϭ Ϫ3, 2,Ϫ1 ,T  1, 1, 1 ϭ 2, 0,Ϫ1 ,

T : R3→  R3