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

By Michael W. Frazier

This article was once initially written for a "Capstone" direction at Michigan nation college. A Capstone path is meant for undergraduate arithmetic majors, as one of many ultimate classes taken of their undergraduate curriculum. Its objective is to compile diversified themes lined within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their purposes. easy wavelet conception appears an ideal subject for this kind of path. As a topic, it dates again purely to 1985. in view that then there was an explosion of wavelet examine, either natural and utilized. Wavelet idea is at the boundary among arithmetic and engineering. particularly it's a reliable subject for demonstrating to scholars that arithmetic study is prospering within the modern-day: scholars can see non-trivial arithmetic rules resulting in average and demanding purposes, similar to video compression and the numerical answer of differential equations. the one necessities assumed are a simple linear algebra heritage and a little bit research history. this article is meant to be as simple an advent to wavelet conception as attainable. it's not meant as a radical or authoritative reference on wavelet conception.

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Example text

Vn are linearly dependent. 9(ii), this implies u ∈ span {v1 , v2 , . . , vn }. Since u is arbitrary, this proves that span {v1 , v2 , . . , vn } V. Now suppose that span {v1 , v2 , . . , vn } V. If v1 , v2 , . . 8(ii), we can find a subset of n − 1 vectors that still spans V. Then w1 , w2 , . . 12. This contradiction shows that v1 , v2 , . . , vn are linearly independent. 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 , .

We say A is invertible if there exists an n × n matrix, denoted A−1 , such that A −1 A I and AA−1 I. We call A−1 , when it exists, the inverse of A. For 2 × 2 matrices, it is worth remembering that if ad − bc a b c d −1 1 ad − bc d −b −c a 0, . Invertible linear transformations correspond to invertible matrices. 4. 55 Let U and V be n-dimensional vector spaces over C (similarly if both are over R). Suppose T : U → V is a linear transformation. Let R be a basis for U and S a basis for V. 49. Then T is an invertible linear transformation if and only if AT is an invertible matrix.

N 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. First, they may be different types of objects. 1(i)), then v is a polynomial whereas [v]S is an n-tuple of numbers. Second, even if the vector space V is Rn or Cn , so that v is an n-tuple of numbers (which is a well-defined object, without reference to any basis), v and [v]S will usually be different n-tuples of numbers (unless S is the Euclidean basis, as we note below).

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