1. Chapter 4 Linear Transformations 4.1 Introduction to Linear Transformations 4.2 The Kernel and Range of a Linear Transformation 4.3 Matrices for Linear Transformations 4.4 Transition Matrices and Similarity

2. 6 - 1 4.1 Introduction to Linear Transformations TV W A linear transformation is a function T that maps a vector space V into another vector space W: V: the domain of T W: the codomain of T Two axioms of linear transformations

3. 6 - 2 Image of v under T: If v is in V and w is in W such that Then w is called the image of v under T. the range of T: the range of T: The set of all images of vectors in V The set of all images of vectors in V. the preimage of w: The set of all v in V such that T(v)=w.

4. 6 - 3 Notes: linear transformationoperation preserving (1) A linear transformation is said to be operation preserving. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W a vector space into itselflinear operator (2) A linear transformation from a vector space into itself is called a linear operator.

5. 6 - 4 Ex: (Verifying a linear transformation T from R 2 into R 2 ) Pf:

6. 6 - 5 Therefore, T is a linear transformation.

7. 6 - 6 Ex: (Functions that are not linear transformations)

8. 6 - 7 Notes: Two uses of the term “linear”. a linear function (1) is called a linear function because its graph is a line. But not a linear transformation because it preserves neither vector addition nor scalar multiplication (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication.

9. 6 - 8 Zero transformation: Identity transformation: Thm 4.1 Thm 4.1: (Properties of linear transformations)

10. 6 - 9 Ex: (Linear transformations and bases) Let be a linear transformation such that Sol: (T is a L.T.) Find T(2, 3, -2).

11. 6 - 10 Ex: (A linear transformation defined by a matrix) The function is defined as Sol: (vector addition) (scalar multiplication)

12. 6 - 11 Thm 4.2 Thm 4.2: (The linear transformation given by a matrix) Let A be an m  n matrix. The function T defined by is a linear transformation from R n into R m. Note:

13. 6 - 12 Show that the L.T. given by the matrix has the property that it rotates every vector in R 2 counterclockwise about the origin through the angle . Rotation in the plane Rotation in the plane Sol: (polar coordinates) r ： the length of v  ： the angle from the positive x-axis counterclockwise to the vector v

14. 6 - 13 r ： the length of T(v)  +  ： the angle from the positive x-axis counterclockwise to the vector T(v) Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle .

15. 6 - 14 is called a projection in R 3. A projection in R 3 A projection in R 3 The linear transformation is given by

16. 6 - 15 Show that T is a linear transformation. A linear transformation from M m  n into M n  m A linear transformation from M m  n into M n  m Sol: Therefore, T is a linear transformation from M m  n into M n  m.

17. 6 - 16 4.2 The Kernel and Range of a Linear Transformation Kernel Kernel of a linear transformation T: Let be a linear transformation kernelker Then the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T). Ex 1: (Finding the kernel of a linear transformation) Sol:

18. 6 - 17 Ex 2: The kernel of the zero and identity transformations: ( a ) T(v)=0 (the zero transformation ) ( b ) T(v)=v (the identity transformation ) Ex 3: (Finding the kernel of a linear transformation) Sol:

19. 6 - 18 Finding the kernel of a linear transformation Finding the kernel of a linear transformation Sol:

20. 6 - 19 Thm 4.3: Thm 4.3: (The kernel is a subspace of V) The kernel of a linear transformation is a subspace of the domain V. Pf: Note: nullspace The kernel of T is also called the nullspace of T.

21. 6 - 20 Finding a basis for the kernel R 5 Find a basis for ker(T) as a subspace of R 5. Sol:

22. 6 - 21 Corollary to Thm 4.3: Corollary to Thm 4.3: Thm 4.4 Thm 4.4: The range of T is a subspace of W Pf:

23. 6 - 22 Notes: Thm 4.4 Corollary to Thm 4.4:

24. 6 - 23 Finding a basis for the range of a linear transformation Find a basis for the range of T. Sol:

25. 6 - 24 Rank of a linear transformation T: V→W: Nullity of a linear transformation T: V→W: Note: Note:

26. 6 - 25 Thm 4.5 Thm 4.5: Sum of rank and nullity Pf:

27. 6 - 26 Finding the rank and nullity of a linear transformation Sol:

28. 6 - 27 One-to-one: One-to-one: one-to-onenot one-to-one

29. 6 - 28 Onto: Onto: i.e., T is onto W when range(T)=W i.e., T is onto W when range(T)=W.

30. 6 - 29 Thm 4.6: Thm 4.6: (One-to-one linear transformation) Pf:

31. 6 - 30 One-to-one and not one-to-one linear transformation One-to-one and not one-to-one linear transformation

32. 6 - 31 Thm 4.7 Thm 4.7: (Onto linear transformation) Thm 4.8 Thm 4.8: (One-to-one and onto linear transformation) Pf:

33. 6 - 32 Ex: Sol: T:Rn→RmT:Rn→Rm dim(domain of T) rank(T)nullity(T)1-1onto 3(a)T:R3→R33(a)T:R3→R3 330YesYes (b)T:R2→R3(b)T:R2→R3 220YesNo 2(c)T:R3→R22(c)T:R3→R2 321 Yes (d)T:R3→R3(d)T:R3→R3 321

34. 6 - 33 Isomorphism IsomorphismPf: Thm 4.9: Thm 4.9: (Isomorphic spaces and dimension) Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension.

35. 6 - 34 Ex: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other.

36. 6 - 35 4.3 Matrices for Linear Transformations Three reasons for matrix representation of a linear transformation: It is simpler to write. It is simpler to read. It is more easily adapted for computer use. Two representations Two representations of the linear transformation T:R 3 →R 3 :

37. 6 - 36 Thm 4.10Standard matrix Thm 4.10: (Standard matrix for a linear transformation)

38. 6 - 37 Pf:

39. 6 - 38

40. 6 - 39 Ex : (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation

41. 6 - 40 Note: Check:

42. 6 - 41 Composition of T 1 : R n →R m with T 2 : R m →R p : Thm 4.11: Thm 4.11: (Composition of linear transformations)

43. 6 - 42 Pf: But note:

44. 6 - 43 Ex : (The standard matrix of a composition) Sol:

45. 6 - 44

46. 6 - 45 Inverse linear transformation Inverse linear transformation Note: If the transformation T is invertible, then the inverse is unique and denoted by T –1.

47. 6 - 46 Existence of an inverse transformation Note: If T is invertible with standard matrix A, then the standard matrix for T –1 is A –1. (1)T is invertible. (2)T is an isomorphism. (3)A is invertible.

48. 6 - 47 Ex : (Finding the inverse of a linear transformation) Sol: Show that T is invertible, and find its inverse.

49. 6 - 48

50. 6 - 49 the matrix of T relative to the bases B and B' Thus, the matrix of T relative to the bases B and B' is

51. 6 - 50 Transformation matrix for nonstandard bases Transformation matrix for nonstandard bases

52. 6 - 51

53. 6 - 52 Ex : (Finding a transformation matrix relative to nonstandard bases) Sol:

54. 6 - 53 Check:

55. 6 - 54 Notes: Notes:

56. 6 - 55 4.4 Transition Matrices and Similarity

57. 6 - 56 Two ways to get from to : Two ways to get from to :

58. 6 - 57 Ex Ex Sol:

59. 6 - 58 with

60. 6 - 59 Similar matrix: Similar matrix: an invertible matrix P For square matrices A and A‘ of order n, A‘ is said to be similar to A if there exist an invertible matrix P such that Thm 4.13: Thm 4.13: (Properties of similar matrices) Let A, B, and C be square matrices of order n. Then the following properties are true. (1) A is similar to A. (2) If A is similar to B, then B is similar to A. (3) If A is similar to B and B is similar to C, then A is similar to C.Pf:

61. 6 - 60 Ex : (A comparison of two matrices for a linear transformation) Sol:

62. 6 - 61