Download presentation

Presentation is loading. Please wait.

Published byMicaela London Modified over 2 years ago

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

Similar presentations

OK

राघव वर्मा Inner Product Spaces Physical properties of vectors aka length and angles in case of arrows Lets use the dot product Length of Cosine of the.

राघव वर्मा Inner Product Spaces Physical properties of vectors aka length and angles in case of arrows Lets use the dot product Length of Cosine of the.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt online viewers Download ppt on pulse code modulation transmitter Ppt on seven segment display pin Ppt on power system stability study Ppt on trans-siberian railway prices Ppt on new technology in banking Ppt on origin of earth Ppt on sikkim cultural Ppt on synthesis and degradation of purines and pyrimidines in dna Ppt on computer languages history