1. Chapter 4 Linear Transformations

2. Outlines Definition and Examples Matrix Representation of linear transformation Similarity

3. Linear transformations are able to describes.  Translation, rotation & reflection  Solvability of  D x &

4. Definition: A mapping L from a vector space V into a vector space W is said to be a linear transformation (or a linear operator) if Remark: L is linear

5. Example 1: Remark: In general, if, the linear transformation can be thought of as a stretching ( ) or shrinking ( ) by a factor of

6. Example 2:

7. Example 3:

8. Example 4:

9. Example 8:

10. Example 9 :

11. Example 10:

12. Lemma:

13. Def:

14. Theorem 4.1.1:

15. Example 11:

16. Example 12:

17. Example 13:

18. Theorem:

19. §4.2 Matrix Representations of Linear Transformations Theorem4.2.1:

20. Proof:

21. Example: Solution:

22. Example: Solution:

23. Figure 4.2.1: (0,1) (1,0) Ax x

26. Theorem4.2.2:

28. Example 3: Solution:

29. Example 4: Solution:

30. Example 5: Solution:

31. Theorem 4.2.3

32. Proof :

33. Cor. 4.2.4: Proof:

34. Example 6 :

35. Solution(Method I):

36. Solution(Method II):

37. Remark:

39. Application I : Computer Graphics and Animation Fundamental operators: Dilations and Contractions: Reflection about : e.g., : a reflection about X-axis. : a reflection about Y-axis.

40. Rotations: Translations: Note: Translation is not linear if Homogeneous Composition of linear mappings is linear!

43. §4.3 Similarity V W L coordinate mapping (transition matrix)

44. Question:

45. Example:

46. Solution:

47. Thm 4.3.1

48. Proof

51. DEFINITION:

52. Remark:

53. Example1:

54. Solution:

55. Example2: Solution: