1. 1 D. R. Wilton ECE Dept. ECE 6382 Introduction to Linear Vector Spaces Reference: D.G. Dudley, “Mathematical Foundations for Electromagnetic Theory,” IEEE Press, 1994.

2. Fields Fields

3. Linear Vector Spaces

4. Linear Vector Spaces, cont’d

5. Field Linear vector space A linear vector space enables us to form linear combinations of vector objects.

6. Linear Vector Space Examples

7. Linear Vector Space Examples, cont’d

8. Linear Independence

9. Dimensionality

10. Linear Independence and Dimensionality

11. Bases Note: If N is finite and dim S = N, then “and if” in the first line above may be replaced by “then”. I.e., any N independent vectors form a basis. Unfortunately, it is not the case that any infinite set of independent vectors forms a basis when dim S = ∞ !

12. Bases, cont’d

16. Inner Product Spaces Field Inner product space The inner product is a generalization of the dot product of vectors in R 3

17. Inner Product Spaces, cont’d

20. Since the inner product generalizes the notion of a dot product of vectors in R 3, we often read as “a dot b” and say that is a “projection of a along b ” or vice versa.

21. The Cauchy-Schwarz-Bunjakowsky (CSB) Inequality

22. The Cauchy-Schwarz-Bunjakowsky (CSB) Inequality, cont’d

23. Orthogonality and Orthonormality

24. Normed Linear Space

25. Normed Linear Space, cont’d

27. Convergence of a Sequence

28. Continuity of the Inner Product

29. Convergence in the Cauchy Sense

30. Convergence in the Cauchy Sense, cont’d

34. Hilbert Spaces

35. Hilbert Spaces, cont’d

36. Linear Subspaces

37. Linear Subspaces, cont’d

38. Gram-Schmidt Orthogonalization

39. Gram-Schmidt Orthogonalization, cont’d

42. Closed Sets

43. Best Approximation in a Hilbert Space

44. Best Approximation in a Hilbert Space, cont’d

50. Orthogonal Complement to a Linear Subspace

51. The Projection Theorem

52. The Projection Theorem and Best Approximation

53. The Projection Theorem and Best Approximation, cont’d

58. Operators in Hilbert Space

59. Operators in Hilbert Space, cont’d

63. Continuity of Hilbert Operators

64. Continuity of Hilbert Operators, cont’d

65. Equivalence of Boundedness and Continuity of Hilbert Operators

66. Unbounded Operator Example

67. Matrix Representation of Bounded Hilbert Operators

68. Matrix Representation of Bounded Hilbert Operators, cont’d

69. Non-Negative, Positive, and Positive Definite Operators

70. Non-Negative, Positive, and Positive Definite Operators, cont’d

72. The Moment Method

73. The Moment Method, cont’d