By Michael W. Frazier

This article was once initially written for a "Capstone" direction at Michigan country college. A Capstone direction is meant for undergraduate arithmetic majors, as one of many ultimate classes taken of their undergraduate curriculum. Its objective is to assemble diversified themes coated within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their purposes. uncomplicated wavelet idea appears an ideal subject for any such path. As a subject matter, it dates again merely to 1985. in view that then there was an explosion of wavelet examine, either natural and utilized. Wavelet conception is at the boundary among arithmetic and engineering. particularly it's a reliable subject for demonstrating to scholars that arithmetic learn is prospering within the modern-day: scholars can see non-trivial arithmetic rules resulting in normal and critical purposes, comparable to video compression and the numerical answer of differential equations. the one must haves assumed are a uncomplicated linear algebra history and a little research historical past. this article is meant to be as common an advent to wavelet thought as attainable. it's not meant as a radical or authoritative reference on wavelet idea.

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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. i. If u ∈ span {v1 , v2 , . . , vn }, prove that u, v1 , v2 , . . , vn are linearly dependent. ii. Suppose v1 , v2 , . . , vn are linearly independent. If u, v1 , v2 , . . , vn are linearly dependent, prove that u ∈ span{v1 , v2 , . . , vn }. 10. Suppose V is an n-dimensional vector space, k ≤ n, and {v1 , v2 , .

The span of U (denoted span U) is the set of all linear combinations of elements of U. In particular, if U is a ﬁnite set, say U {u1 , u2 , . . un }, then span U n j 1 αj uj : αj ∈ F for all j 1, 2, . . , n . 3. Vector Spaces and Bases 33 To visualize the span, note that the span of a single nonzero vector u in Rn consists of all vectors lying on the line through the origin that contains u. If u and v are two noncollinear vectors in R3 , then span {u, v} is the plane through the origin containing u and v.

I. (Factor theorem) Let P be a polynomial and let a ∈ C. We say z − a divides P, or z − a is a factor of P (written (z − a)|P), if there exists a polynomial Q such that P(z) (z − a)Q (z). Prove that (z − a)|P if and only if P(a) 0; that is, z − a is a factor of P if and only if a is a root of P. Hint: The “only if” direction is immediate. For the “if” direction, the result follows easily from the division algorithm for polynomials, if you are familiar with that. If not, a more elementary proof can be obtained by ﬁrst showing that (z − a)|(z k − ak ) for any integer k ≥ 1.