 33.33.1: Write each of the following complex numbers in the form a + bi. The...
 33.33.2: Write each of the following complex numbers in the form a + bi. The...
 33.33.3: Write each of the following complex numbers in the form a + bi. The...
 33.33.4: Write each of the following complex numbers in the form a + bi. The...
 33.33.5: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.6: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.7: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.8: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.9: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.10: Use DeM oivre' s Theorem to write each of the following numbers in ...
 33.33.11: Determine all complex eighth roots of unity, and represent them geo...
 33.33.12: Determine all complex fifth roots of unity, and represent them geom...
 33.33.13: Express each of the following complex numbers in polar form. 1 + i
 33.33.14: Express each of the following complex numbers in polar form. v'3  i
 33.33.15: Express each of the following complex numbers in polar form. 5
 33.33.16: Express each of the following complex numbers in polar form. 2i
 33.33.17: Express each of the following complex numbers in polar form. 2  2i
 33.33.18: Express each of the following complex numbers in polar form. 2i + 2v'3
 33.33.19: Prove that if z = r(cos e + i sin e) and z "10, thenZI = rI[cos(...
 33.33.20: State and prove DeMoivre's Theorem for negative integers. (For n = ...
 33.33.21: Prove that if n is a positive integer, then the set of all nth root...
 33.33.22: Prove that for each integer n > I the sum of the nth roots of unity...
 33.33.23: Let n be a positive integer. What is the product of all the nth roo...
 33.33.24: Prove that the set of all roots of unity forms a group with respect...
 33.33.25: Prove that with respect to multiplication, the set of all complex n...
 33.33.26: Let z* denote the conjugate of the complex number z, that is, (a + ...
 33.33.27: Prove that for each integer n ::: I, the n complex nth roots ofarez...
 33.33.28: Prove that for each integer n ::: I, the n complex nth roots ofz = ...
 33.33.29: Use 33.27 to find the complex cube roots of 2i.
 33.33.30: Use 33.28 to find the complex cube roots of 2i.
 33.33.31: Use 33.27 to find the complex fourth roots of 5.
 33.33.32: Use 33.28 to find the complex fourth roots of 5.
 33.33.33: Let rc# denote the multiplicative group of nonzero complex numbers ...
 33.33.34: Verify that a : lR + rc defined by a(x) = cos(2Jrx) + i sin(2Jrx) ...
Solutions for Chapter 33: COMPLEX ROOTS OF UNITY
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 33: COMPLEX ROOTS OF UNITY
Get Full SolutionsModern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Chapter 33: COMPLEX ROOTS OF UNITY includes 34 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 34 problems in chapter 33: COMPLEX ROOTS OF UNITY have been answered, more than 8968 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).