By Michael W. Frazier

Mathematics majors at Michigan kingdom collage take a "Capstone" direction close to the top in their undergraduate careers. The content material of this direction varies with each one delivering. Its function is to collect assorted issues from the undergraduate curriculum and introduce scholars to a constructing zone in arithmetic. this article was once initially written for a Capstone direction. simple wavelet thought is a ordinary subject for this sort of path. via identify, wavelets date again basically to the Eighties. at the boundary among arithmetic and engineering, wavelet conception exhibits scholars that arithmetic examine continues to be thriving, with vital purposes in components resembling snapshot compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet idea are sufficiently user-friendly to learn effectively to complicated undergraduates. this article is meant for undergraduates, so just a simple heritage in linear algebra and research is thought. we don't require familiarity with advanced numbers and the roots of unity.

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It implies that any polynomial factors completely over C. 29 Let P be a polynomial of degree N ~ 1, with leading coefficient aN. Then there exist complex numbers z 1 , z2 , • . 16. 1. 2. 3. 4. 14. 15. 16. A series of complex numbers is conditionally convergent if it is convergent but not absolutely convergent. Let 'L~ 1 ak be a conditionally convergent series of real numbers. Let a be an arbitrary real number. , a 1-1, onto map from N to N) such that the reindexed series L~ 1 arr(k) converges to a.

Suppose Vis a finite dimensional vector space over C with basis R. Let v1 , v2 , ... , Vm be elements of V. Prove that the set {v 1 , v 2 , ... , Vm} is linearly independent in V if and only if the set {[v1]R, [Vz]R, ... [vm]R} is linearly independent in Vn Pn- 1 , I en. 40 1. 4 Linear 'Iransformations, Matrices, and Change of Basis In mathematics we often consider the class of all objects having a certain structure, for example, groups, fields, metric spaces, or topological spaces. In each case we consider maps between these objects that are consistent with or preserve this structure, such as homomorphisms or isomorphisms of groups, isometries of metric spaces, and homeomorphisms of topological spaces.

5. 5) is a useful formula that we apply for other purposes in chapter 2. We now consider power series. 2. 20 series of the form E 19 C. A power series about z0 is a L an(z - zo)n, 00 where an n=O