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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.

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Fields Fields

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Linear Vector Spaces

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Linear Vector Spaces, cont’d

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

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Linear Vector Space Examples

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Linear Vector Space Examples, cont’d

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Linear Independence

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Dimensionality

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Linear Independence and Dimensionality

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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 = ∞ !

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Bases, cont’d

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

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Inner Product Spaces, cont’d

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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.

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The Cauchy-Schwarz-Bunjakowsky (CSB) Inequality

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

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Orthogonality and Orthonormality

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Normed Linear Space

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Normed Linear Space, cont’d

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Convergence of a Sequence

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Continuity of the Inner Product

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Convergence in the Cauchy Sense

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Convergence in the Cauchy Sense, cont’d

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Hilbert Spaces

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Hilbert Spaces, cont’d

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Linear Subspaces

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Linear Subspaces, cont’d

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Gram-Schmidt Orthogonalization

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Gram-Schmidt Orthogonalization, cont’d

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Closed Sets

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Best Approximation in a Hilbert Space

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Best Approximation in a Hilbert Space, cont’d

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Orthogonal Complement to a Linear Subspace

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The Projection Theorem

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The Projection Theorem and Best Approximation

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The Projection Theorem and Best Approximation, cont’d

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Operators in Hilbert Space

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Operators in Hilbert Space, cont’d

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Continuity of Hilbert Operators

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Continuity of Hilbert Operators, cont’d

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Equivalence of Boundedness and Continuity of Hilbert Operators

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Unbounded Operator Example

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Matrix Representation of Bounded Hilbert Operators

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Matrix Representation of Bounded Hilbert Operators, cont’d

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Non-Negative, Positive, and Positive Definite Operators

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

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The Moment Method

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The Moment Method, cont’d

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