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D. R. Wilton ECE Dept.

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

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D. R. Wilton ECE Dept.

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

  2. Fields

  3. Linear Vector Spaces

  4. Linear Vector Spaces, cont’d

  5. Linear Vector Spaces, cont’d 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 dimS =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 dimS =∞!

  12. Bases, cont’d

  13. Bases, cont’d

  14. Bases, cont’d

  15. Bases, cont’d

  16. Field Inner Product Spaces Inner product space The inner product is a generalization of the dot product of vectors in R3

  17. Inner Product Spaces, cont’d

  18. Inner Product Spaces, cont’d

  19. Inner Product Spaces, cont’d

  20. Inner Product Spaces, cont’d Since the inner product generalizes the notion of a dot product of vectors in R3, we often read <a,b> as “a dot b” and say that <a,b> 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

  26. 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

  31. Convergence in the Cauchy Sense, cont’d

  32. Convergence in the Cauchy Sense, cont’d

  33. 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

  40. Gram-Schmidt Orthogonalization, cont’d

  41. Gram-Schmidt Orthogonalization, cont’d

  42. Closed Sets

  43. Best Approximation in a Hilbert Space

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

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

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

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

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

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

  50. Orthogonal Complement to a Linear Subspace

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