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

By Michael W. Frazier

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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!