1. CHAPTER Four Linear Transformations

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

3. Linear transformations are able to describes. - translation, rotation & reflection - linear dynamics - solvability of -

4. Definition: A mapping L from a vector space into another vector space is said to be linear if

5. Example: is linear ( 每一向量放大 3 倍但方向不變 ) Example: ( 投影到 一軸 ) Example: ( 對 軸反射 )

6. Example: ( 對 線反射 )( 用 Householder transformation 證明 ) Example: ( 旋轉 角度 ) Example:

8. Lemma: Let be a linear transformation. Then (i) (ii) (iii) Pf: (i) Let

9. Def: Let be a linear transformation. Then is called kernel of L. On the other hand, if S is a subspace of, then is called the image of S. If, then is the range of L.

10. Theorem 4.1.1: Let be a linear transformation, and S is a subspace of. Then (i) (ii) Pf: trivial

11. Example: Then Example: Note that

12. Example:

13. Theorem: Let be a linear transformation. Then L is an injection pf: L is one-to-one

14. Observation: Let be linear Let This leads to the next results. Theorem 4.2.1: Let be linear Then Where

15. Example: Let We compute Then

16. Example: Determine a linear mapping from which rotates each vector by angle in the counterclockwise direction. Sol: Thus, the direction linear transform is

17. Question: Is there a matrix representation for a linear mapping L from a general vector space ( other than ) into another vector space? Yes! In the sense of Question: How to find ? ( 其精神在於基底映射 )

18. Let and Then where

19. Note: Let be two ordered basis for Recall that the transition matrix from to is determined from Thus can be interpreted as the matrix representation of

20. Example: Let Find with respect to Sol: Thus Check: i.e.,

21. Example: Let be an ordered basis for If is selected as ordered basis both for domain and range Then since

22. Example: Find with respect to basis Sol: Check:

23. Question: Let be linear. Can be computed more efficient?

24. Let This leads to the next result Theorem: can be computed by Gaussian elimination from

25. Example: Let Find A L with respect to

26. Change of basis: Let be two ordered basis Then Matrix Representation: where Transition matrix in change of basis is a special case of Matrix representation with

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

28. Rotations: Translations: Note: Translation is not linear if Homogeneous Composition of linear mapping is linear!

29. Similarity Similarly, Question: What is the relation between Question: How to choose basis such that is as simple as possible or is in the desired form?

30. Example: Let Question: How are related? From (*) and (**),

31. Theorem4.3.1: Where is the transition matrix from ordered basis F to E.

32. Pf: Let Similarity,

33. Def: Let B is said to be similar to A, denoted by B~A, if nonsingular matrix S such that Question: Do you find the relation between similarity of two matrices & change of basis for a linear mapping? Question: How to choose S such that is as simple as possible? Question: Have you noticed the relation between S & the ordered basis?

34. Example: Let D be the differential operator, By direct calculation which agree with theoretical result.

35. Example: Let. Find the matrix representing L with respect to where Sol: Method 1:

36. Method 2: