 14.14.1: Solve the equation (1 2)x = (1 2 3) in 53'
 14.14.2: Solve the equation x(l 3 2) = (1 3) in 53.
 14.14.3: Detennine the elements in each of the cyclic subgroups of 53. Also ...
 14.14.4: Detennine the elements in each of the cyclic subgroups of Z6. Also ...
 14.14.5: Find the order of the element (1 2)(3 4) in 54. Verify that {(1), (...
 14.14.6: Find the order of the element (1 2)(3 4 5) in 55.
 14.14.7: (a) Detennine the elements in the subgroup (1 2 3 4)} of 54.(b) Det...
 14.14.8: Detennine the elements in each of the following subgroups of the gr...
 14.14.9: Let a denote the clockwise rotation of the plane through 90 about a...
 14.14.10: (a) Repeat 14.9 with 40 in place of 90.(b) What is the order of (a)...
 14.14.11: Prove Theorem 14.1(b).
 14.14.12: Prove that xa = b has a unique solution in a group. This is the omi...
 14.14.13: Prove that axb = c has a unique solution in a group (given a, b, c).
 14.14.14: (a) Prove that if a and b are elements of an Abelian group G, with ...
 14.14.15: Show with an example that if G is not Abelian, then the statement i...
 14.14.16: (a) Use 14.14 to prove that in an Abelian group the elements of fin...
 14.14.17: Verify Theorem 14.1(e) for a = (1 2 5) and b = (2 3 4) in S5. Is (a...
 14.14.18: Assume that a and b are elements of a group G.(a) Provethatab = ba ...
 14.14.19: Assume m, n E Z. Find necessary and sufficient conditions for (m) S...
 14.14.20: Construct a Cayley table for a group G given that G = (a), a I e, ...
 14.14.21: Rewrite Theorem 14.1 (not its proof) for a group written additively...
 14.14.22: (a) Prove that if a, b, and e are elements of a group, then anyone ...
 14.14.23: Prove that a nonidentity element of a group has order 2 iff it is i...
 14.14.24: Prove that every group of even order has an element of order 2. ( 1...
 14.14.25: Prove that a group G is Abelian iff (ab)l = a1b 1 for all a, bEG.
 14.14.26: There is only one way to complete the following Cayley table so as ...
 14.14.27: Assume that (x, y, z, wj is to be a group, with identity x (operati...
 14.14.28: Prove that if a is a fixed element of a group G, and)" : G + G is ...
 14.14.29: Prove that a group is Abelian if each of its nonidentity elements h...
 14.14.30: Prove that if G is a group and a E G, then o(a 1 ) = o(a).
 14.14.31: Prove that if G is a group and a, bEG, then 0(a1ba) = o(b).
 14.14.32: Prove that if G is a group and a, bEG, then o(ab) = o(ba). (Suggest...
 14.14.33: Prove or give a counterexample: If a group G has a subgroup of orde...
 14.14.34: Prove that if a group G has no subgroup other than G and (e), then ...
 14.14.35: Prove that if G is a finite group, then Theorem 7.1 is true with co...
 14.14.36: Prove that the order of an element Ci in S" is the least common mul...
 14.14.37: Determine the largest order of an element of S" for each n such tha...
 14.14.38: Prove that if A and B are subgroups of a group G, and A U B is also...
Solutions for Chapter 14: ELEMENTARY PROPERTIES
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 14: ELEMENTARY PROPERTIES
Get Full SolutionsThis textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 38 problems in chapter 14: ELEMENTARY PROPERTIES have been answered, more than 8967 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Chapter 14: ELEMENTARY PROPERTIES includes 38 full stepbystep solutions.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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 variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.