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By Michael W. Frazier

Mathematics majors at Michigan kingdom college take a "Capstone" path close to the top in their undergraduate careers. The content material of this path varies with each one delivering. Its goal is to compile varied subject matters from the undergraduate curriculum and introduce scholars to a constructing zone in arithmetic. this article used to be initially written for a Capstone direction. easy wavelet idea is a average subject for this sort of direction. by way of identify, wavelets date again in basic terms to the Eighties. at the boundary among arithmetic and engineering, wavelet idea indicates scholars that arithmetic examine continues to be thriving, with very important purposes in parts comparable to photograph compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet concept are sufficiently undemanding to study effectively to complex undergraduates. this article is meant for undergraduates, so just a simple historical past in linear algebra and research is believed. we don't require familiarity with complicated numbers and the roots of unity.

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Use part i to obtain 0 < -1. 5. 7. 6. 8. Suggestion: Do not write it out in terms of the real and imaginary parts. 7. 7. For z, wE C, define d(z, w) = jz- wj. 2 for the definition). 2 is called the triangle inequality. 8. 2) is a field by checking properties Al-AS, Ml-MS, and D. 9. 9. 10. Let {zn}~M be a sequence of complex numbers. For each n, let Zn = Xn + iyn, where Xn,Yn E JR. 16 1. Background: Complex Numbers and Linear Algebra i. 4). ii. 3). iii. 12. 2 Complex Series, Euler's Formula, and the Roots of Unity We begin with series of complex numbers.

Clearly I is linear. 49. 49. 58 the R to S change of basis matrix. 59 Define bases R and S for JR 2 by and S={[~J. ]}· Find the R to S change of basis matrix A. 58, we need to find {aij} such that [ ~2 J= an [ ~ J+ a21 [ ! J, and Solving these linear systems gives an=2, a 12 = -5, a21=-l, and a22 = 3. SO 1. Background: Complex Numbers and Linear Algebra Hence 2 A= [ -1 -5 3 J. 14). Now we restrict our attention to a linear transformation from a finite dimensional vector space to itself. Fix a basis and consider representing the linear transformation in that basis.

6) continue to hold in the complex case. For example, relation (1. 8) for all z E C. 7) leads to Euler's formula, a very elegant and useful identity. 22 E C, eiz = cosz + i sinz. 9) Proof We apply relation (1. 7) with iz in place of z and collect the even and odd powers of z: iz . (izi (izi (iz) 4 (iz) 5 (iz) 6 e =1+tz+-+-+-+-+-+oo• i2z2 2 3! i4z4 4! i6z6 5! 6! i3z3 iszs = (1 + 2 + 4! + 6! + · · ·) + (iz + 3! + 5! + .. ·) z4 z2 = (1 - 2 + 4! z6 - 6! = cosz + i sinz. + .. ·) + i(z- z3 3! zs + 5!

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