An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier

By Michael W. Frazier

This article used to be initially written for a "Capstone" path at Michigan nation collage. A Capstone direction is meant for undergraduate arithmetic majors, as one of many ultimate classes taken of their undergraduate curriculum. Its function is to assemble diverse issues lined within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their purposes. easy wavelet idea appears to be like an ideal subject for this kind of path. As a subject matter, it dates again simply to 1985. seeing that then there was an explosion of wavelet learn, either natural and utilized. Wavelet conception is at the boundary among arithmetic and engineering. particularly it's a stable subject for demonstrating to scholars that arithmetic study is prospering within the modern-day: scholars can see non-trivial arithmetic rules resulting in usual and demanding purposes, equivalent to video compression and the numerical answer of differential equations. the single must haves assumed are a uncomplicated linear algebra historical past and slightly research history. this article is meant to be as undemanding an creation to wavelet conception as attainable. it's not meant as a radical or authoritative reference on wavelet concept.

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Then T is determined by its action on the basis {u1 , u2 , . . , un }, in the following sense. Suppose u ∈ U. Then there exist unique scalars α1 , α2 , . . 38. By properties L1 and L2, T(u) T n j 1 α j uj n αj T(uj ). j 1 In particular, suppose L : U → V is a linear transformation, and L(uj ) T(uj ) for all j 1, 2, . . , n. Then L T, that is, L(u) T(u) for all u ∈ U, because the steps above also show that n L(u) j 1 αj L(uj ). 43) that for a given basis S {u1 , u2 , . . , Rn or Cn ) whose components α1 , α2 , .

43 Suppose V is a vector space over a field F and S {v1 , v2 , . . , vn } is a basis for V. For any vector v ∈ V, there exist n unique α1 , α2 , . . 38. We n denote by [v]S the vector in F with components α1 , α2 , . . , αn , that is,   α1  α2    [v]S  .  . 23)  ..  αn We call αj the jth component of v with respect to S. 36 1. Background: Complex Numbers and Linear Algebra In other words, for a basis S {v1 , v2 , . . 23) n means that v j 1 αj v j . One should not confuse [v]S with the vector v itself.

12. 2 Complex Series, Euler’s Formula, and the Roots of Unity We begin with series of complex numbers. Particular cases of interest are geometric series and the power series for sin z, cos z, and ez . Using these we establish Euler’s formula eiθ cos θ + i sin θ. This will lead to the polar representation of complex numbers and allow us to calculate N th roots of complex numbers, especially the N th “roots of unity,” the roots of the number 1. In chapter 2 we write Fourier expansions of vectors using the complex exponentials introduced here.

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