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 T VW 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 co-domain 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 pre-image 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 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:

12. 6 - 11 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

13. 6 - 12 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 .

14. 6 - 13 is called a projection in R 3. A projection in R 3 A projection in R 3 The linear transformation is given by

15. 6 - 14 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.

16. 6 - 15 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).

17. 6 - 16 Finding the kernel of a linear transformation Finding the kernel of a linear transformation Sol:

18. 6 - 17 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: Corollary to Thm 4.3: Corollary to Thm 4.3:

19. 6 - 18 Finding a basis for the kernel R 5 Find a basis for ker(T) as a subspace of R 5. Sol:

20. 6 - 19 Thm 4.4 Thm 4.4: The range of T is a subspace of W Pf:

21. 6 - 20 Rank of a linear transformation T: V→W: Nullity of a linear transformation T: V→W: Note: Note:

22. 6 - 21 Finding a basis for the range of a linear transformation Find a basis for the range(T). Sol:

23. 6 - 22 Thm 4.5 Thm 4.5: Sum of rank and nullity Pf:

24. 6 - 23 Finding the rank and nullity of a linear transformation Sol:

25. 6 - 24 One-to-one: One-to-one: one-to-onenot one-to-one

26. 6 - 25 Onto: Onto: range(T)=W i.e., T is onto W when range(T)=W.

27. 6 - 26 Thm 4.6: Thm 4.6: (One-to-one linear transformation) Pf:

28. 6 - 27 One-to-one and not one-to-one linear transformation One-to-one and not one-to-one linear transformation

29. 6 - 28 Onto linear transformation Onto linear transformation Thm 4.7 Thm 4.7: (One-to-one and onto linear transformation) Pf: Note:

30. 6 - 29 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 Note:

31. 6 - 30 Isomorphism IsomorphismPf: Thm 4.8: Thm 4.8: (Isomorphic spaces and dimension) Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension.

32. 6 - 31 Ex: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other.

33. 6 - 32 4.3 Matrices for Linear Transformations matrix representation 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 :

34. 6 - 33 Thm 4.9Standard matrix Thm 4.9: (Standard matrix for a linear transformation)

35. 6 - 34 Pf:

36. 6 - 35

37. 6 - 36 Ex : (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation

38. 6 - 37 Note: Check:

39. 6 - 38 Composition of T 1 : R n →R m with T 2 : R m →R p : Thm 4.10: Thm 4.10: (Composition of linear transformations)

40. 6 - 39 Pf: But note:

41. 6 - 40 Ex : (The standard matrix of a composition) Sol:

42. 6 - 41

43. 6 - 42 Inverse linear transformation Inverse linear transformation Note: If the transformation T is invertible, then the inverse is unique and denoted by T –1.

44. 6 - 43 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.

45. 6 - 44 Ex : (Finding the inverse of a linear transformation) Sol: Show that T is invertible, and find its inverse.

46. 6 - 45

47. 6 - 46 the matrix of T relative to the bases B and B' Thus, the matrix of T relative to the bases B and B' is

48. 6 - 47 Transformation matrix for nonstandard bases Transformation matrix for nonstandard bases

49. 6 - 48

50. 6 - 49 Ex : (Finding a transformation matrix relative to nonstandard bases) Sol:

51. 6 - 50 Check:

52. 6 - 51 Notes: Notes:

53. 6 - 52 4.4 Transition Matrices and Similarity

54. 6 - 53 Two ways to get from to : Two ways to get from to :

55. 6 - 54 Ex Ex Sol:

56. 6 - 55 with

57. 6 - 56 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 4.12: Thm 4.12: (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:

58. 6 - 57 Ex : (A comparison of two matrices for a linear transformation) Sol:

59. 6 - 58