By Michael W. Frazier

ISBN-10: 3642855709

ISBN-13: 9783642855702

ISBN-10: 3642855725

ISBN-13: 9783642855726

Mathematics majors at Michigan country college take a "Capstone" direction close to the top in their undergraduate careers. The content material of this direction varies with each one supplying. Its objective is to assemble assorted subject matters from the undergraduate curriculum and introduce scholars to a constructing quarter in arithmetic. this article used to be initially written for a Capstone direction. simple wavelet concept is a average subject for the sort of path. by means of identify, wavelets date again merely to the Nineteen Eighties. at the boundary among arithmetic and engineering, wavelet concept indicates scholars that arithmetic learn continues to be thriving, with vital purposes in parts akin to picture compression and the numerical answer of differential equations. the writer believes that the necessities of wavelet concept are sufficiently basic to study effectively to complex undergraduates. this article is meant for undergraduates, so just a easy history in linear algebra and research is believed. we don't require familiarity with complicated numbers and the roots of unity.

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**Example text**

A vector) x. The result y =Ax is an m x 1 matrix, that is, a vector with m components. 2). 47, the ith component of Ax is 11 (Ax)i = L:aikXk. k=l For more general finite dimensional vector spaces, a matrix leads to a linear transformation by acting on the vector of components with respect to a given basis as follows. 48 (Linear transformation associated with a matrix) Let U and V be finite dimensional vector spaces over a field JF. Suppose R = {u 1 , u 2 , ... , u 11 } is a basis for U andS = {v1, v2, ...

Bases are also characterized in the following way. 38 Let V be a vector space over a field lF, and let U be a nonempty subset of V. i. Suppose U is finite, say U {u 1 , u2 , •.. , Un}, for some n E N, with ui =f:. uk for j =f:. k. Then U is a basis for V if and only if = 34 1. •• , an e lF such that LJ=l v = ajUj. ii. If U is infinite, then U is a basis for V if and only if for each non-zero v e V, there exist unique m e N, u1 , u 2, ... , Urn e U, and nonzero a1, a2, ... , am· e lF such that v = 'Lj:1 aiui.

25). Proof Let u e U be arbitrary. •• , Cn be the components ofu with respect toR, that is, [u~= or, equivalently, u By linearity, Hence the ith [ ll = Ljn=I CjUj· By definition of A, T(uj) = L:;-: 1 0-ijVi. component of T(u) with respect to the basis S is L:l=I OijCj, which is, by definition, the ith component of the product 44 1. 25) holds. 6). • In the previous lemma, we call A the matrix representing T with respect toR and S. We sometimes write it as Ar. 50 Define a basis R for JR2 by R={[~J.

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