1. Chapter 3Vector Space • Objective: To introduce the notion of vector space, subspace, linear independence, basis, coordinate, and change of coordinate.

2. §3-1 Definition and Examples Recall, In - vector addition - scalar multiplication - norm - triangle inequality

3. m Why Introduces Vector Space? • It provides comprehensive understanding of many mathematical & physical phenomena. For example, • All the solutions of the ODE can be described as . Why? • Controllability and observability space in linear control theory.

4. Vector Space Axioms Definition: Let be set and be a field ( in most practical case, ). Define two binary operations Then is a vector space if the follow- ing Conditions hold:

5. Vector Space Axioms (cont.) For any , A1: A2: A3: A4:

6. Vector Space Axioms (cont.) A5: A6: A7:

7. Examples • defined by • over is a vector space.

8. Examples (cont.) • over is also a vector space with defined by (1) and (2). • over is a vector space. • over is NOT a vector space. (Why?)

9. Examples (cont.) • Let • over defined by is a vector space.

10. Examples (cont.) • defined by is a vector space. • is NOT a vector space. (Why?) • is NOT a vector space. (Why?)

11. Theroem3.1.1: Let be a vector space and . Then PF:

12. §3-2 Subspace Definition: If is a nonempty subset of a vector space , and satisfies the following conditions: then is said to be a subspace of . Remark 1: Thus every subspace is a vector space in its own right. Remark 2: In a vector space , it can be readily verified that and are subspaces of . All other subspaces are referred to as propersubspaces.

13. Examples of Subspaces Example 2.(P.135)

14. Examples of Subspaces (cont.) Example 3.(P.135) Example 4.(P.135)

15. Examples of Subspaces (cont.) Example 5.(P.136) Example 6.(P.136) Example 8.(P.136)

16. Nullspace and Range-space • Let , ※ Define that N(A) is called the nullspace of A; R(A) is called the range(column) space of A.

17. Examples of Nullspaces Example 9.(P.137) Question:Determine N(A) if . Answer:

18. Note Note that, both the vector spaces and the solution set of contain infinite number of elements. Question:Can a vector space be described by a set of vectors with number being as small as possible? Example: Spanning set, linear independent, basis

19. Span and Spanning Sets Definition: Let be vectors in a vector space , a sum of the form , where are scalars, is called linearcombination of . Definition: Definition: is said to be a spanningset for if

20. Examples of Span Example :

21. Theroem3.2.1:If , then is a subspace of . Question:Given a vector space and a set , how to determine whether or not?

22. Example 11. (P.140) Yes, Yes, let ∵ A is nonsingular, The system has a unique solution

23. Example 11.(c) (P.141) No,

24. Example 12. (P.141) Yes, let

25. §3-3 Linear Independence Question: How to find a minimalspanning set of a vector space (i.e. a spanning set that contains the smallest possible number of vectors.) (i.e. There is no redundancy in a spanning set.) It’s unnecessary.

26. Linear Dependency Definition: is said to be linearindependent if “ ”. Definition: is said to be lineardependent if there exist scalars NOT all zero such that

27. Lemma :

28. Note 1:Linear independency means there is no redundancy on the spanning set . Note 2: is a minimal spanning set for iff is linear independent and spans . Definition: A minimal spanning set is called a basis.

29. Linear Dependency (cont.) Question: How to systematically determine the linear dependency of vectors ? • Geometrical interpretation(see Figure 3.3):

30. Example 3. (P.149) • Note that is redundant for the spanning set. • On the other hand, ∵ A is singular det(A)=0. a nontrivial solution is linear dependent. Th 1.4.3

31. Theroem3.3.1: Let , Then is linear independent PF:

32. Example 4. (P.150)

33. Theroem3.3.2:Suppose Then PF:

34. How to determine linear independency For the Vector Space Pn(P.151) Question:Determine the linear dependency of Sol:

35. How to determine linear independency For the Vector Space C(n-1)[a,b](P.152) Let Suppose

36. Wronskian Definition: Let be functions in C(n-1)[a,b], and define thus, the function is called the Wronskian of

37. Theroem3.3.3: Let if are linear dependent on [a,b] Cor:

38. Example of Wronskian Example 6.(P.153) • Is linear independent in Yes, Example 8.(P.154) • Is linear independent in Yes,

39. Question: Does the converse of Th 3.3.3 hold? Answer: No, a counterexample is given as follows Question: Is linear independent in and Why?

40. §3-4 Basis and Dimension Definition:Let be a basis for a vector space if Example:It is easy to show that

41. Theroem3.4.1:Suppose PF:

42. Cor:If are both bases for a vector , then PF:

43. Dimension Definition: Let be a vector space. If has a basis consisting of n vectors, we say that has dimensionn. • { } is said to have dimension 0. • is said to be finite dimensional if finite set of vectors that spans ;otherwise we say is infinite-dimensional.

44. Example of Dimension Example

45. Theroem3.4.3:If , then linear independent PF:

46. Theroem3.4.4:

47. Standard Basis

48. §3-5 Chang of Basis • 不同場合用不同座標系統有不同的方便性，如質點 運動適合用體座標 (body frame) 來描述，而飛彈攔 截適合用球面座標。 • 利用某些特定基底表示時，有時更易使系統特性彰 顯出來。 Question: 不同座標系統間如何轉換？

49. Definition:Let be a vector space and let be an ordered basis for . unique expression

50. Remark 1: Lemma 2: Every n-dimensional vector space is isomorphic to