By Michael W. Frazier

ISBN-10: 3642855709

ISBN-13: 9783642855702

ISBN-10: 3642855725

ISBN-13: 9783642855726

Mathematics majors at Michigan nation college take a "Capstone" path close to the top in their undergraduate careers. The content material of this direction varies with each one supplying. Its objective is to compile various issues from the undergraduate curriculum and introduce scholars to a constructing region in arithmetic. this article was once initially written for a Capstone path. uncomplicated wavelet conception is a ordinary subject for this sort of direction. by way of identify, wavelets date again purely to the Eighties. at the boundary among arithmetic and engineering, wavelet thought exhibits scholars that arithmetic examine continues to be thriving, with vital functions in parts akin to photograph compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet conception are sufficiently easy to study effectively to complicated undergraduates. this article is meant for undergraduates, so just a easy historical past in linear algebra and research is thought. we don't require familiarity with complicated numbers and the roots of unity.

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**Additional resources for An Introduction to Wavelets Through Linear Algebra**

**Sample text**

N. Writing uk in terms of v 1 , ... , Vn, show that the terms cancel out. Hence Uk E span{v1, ... , Vn-d· By the induction assumption, u 1 , u 2 , ... , Un are linearly dependent. Show that this implies that w 1 , . , Wn+ 1 are linearly dependent. 13. 40. 12. 14. Let R be a basis for a finite dimensional vector space V over a field IF. Prove that: i. [u + v]R = [u]R + [v]R for all u, v E V; ii. [av]R = a[v]R for all a E IF and v E V. 15. Suppose Vis a finite dimensional vector space over C with basis R.

56. 55. Hence range T = en. So range T has dimension n, that is, rank A = n. Conversely, if rank A = n, then range T has dimension n. This implies range T = en. (Proof: Let z E en be arbitrary. Then the basis elements u1 , u2 , ... 12. 9(ii), z E span{u 1 , u2 , ... ) SoT is onto. 8(v), Tis 1 - 1, hence invertible. 55, A is invertible. , replace e everywhere by JR. • I Having learned a few prerequisites about matrices, we are ready to return to our study of representing vectors in different bases.

Suppose Vis ann-dimensional vector space over JR. or C. Suppose R = {u1 , u 2 , ••• , Un} and S = {v 1 , Vz, ... , Vn} are two bases for V. Since Sis a basis for V, there are unique scalars aij, i = 1, 2, ... , n, and}= 1, 2, ... Vn, Uz = a12V1 + azzVz + · · · + anzVn, u1 = anVI l ::: :: Let A be the n x n matrix whose kth column consists of the coefficients alk, azk, ... , ank in the representation ofuk: A= [=~~ =~~ a~1 a~z a1n . 4. Linear Transformations, Matrices, and Change of Basis 49 Then [x]s = A[x]R for all x E V, and A is the unique matrix with this property.

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