By Michael W. Frazier

This article was once 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 objective is to collect diversified subject matters coated within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their purposes. uncomplicated wavelet idea looks an ideal subject for this kind of path. As an issue, it dates again merely to 1985. due to the fact that then there was an explosion of wavelet study, either natural and utilized. Wavelet thought is at the boundary among arithmetic and engineering. specifically 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 common and significant functions, corresponding to video compression and the numerical answer of differential equations. the one necessities assumed are a simple linear algebra historical past and a little research heritage. this article is meant to be as ordinary an advent to wavelet conception as attainable. it isn't meant as an intensive or authoritative reference on wavelet thought.

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Nondegeneracy) d(x, y) 0 if and only if x y; Me3. (Metric triangle inequality) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. A metric space (X, d) is a set X with a metric d. For x, y ∈ R, deﬁne d(x, y) |x − y|. Prove that d is a metric on R. 3. Prove that a convergent sequence {xn }∞ n M of real numbers is a Cauchy sequence. 4. Let F with the relation < be an ordered ﬁeld. i. Suppose x ∈ F and x 0. Prove that x2 > 0. ii. Prove that there is no ordering < on the ﬁeld C that makes C an ordered ﬁeld.

7. Prove: if U is a ﬁnite collection of vectors that has a linearly dependent subset, then U is linearly dependent. 8. Suppose v1 , v2 , . . , vn are linearly dependent vectors in some vector space. i. Prove that there is some j ∈ {1, 2, 3, . . , n} such that vj ∈ span{v1 , v2 , . . , vj−1 , vj+1 , . . , vn }. ii. For j as in part i, prove that span{v1 , v2 , . . , vj , . . , vn } span{v1 , v2 , . . , vj−1 , vj+1 , . . , vn }. 9. Suppose u, v1 , v2 , . . , vn are vectors in some vector space.

Matrix multiplication is not commutative: ﬁrst, when AB is deﬁned, it may be that BA is not even deﬁned; second, even if both are 42 1. Background: Complex Numbers and Linear Algebra deﬁned, we may have AB BA. However, matrix multiplication is associative: for A [aij ]1≤i≤m,1≤j≤ , B [bij ]1≤i≤ ,1≤j≤k , C [cij ]1≤i≤k,1≤j≤n , the (i, j)th entry of the m × n matrices (AB)C and A(BC) are both k aip bpq cqj . , a vector) x. The result y Ax is an m × 1 matrix, that is, a vector with m components. 2).