- 18.18.1: Prove that Z is isomorphic to the multiplicative group of all ratio...
- 18.18.2: Prove that Z x Z is isomorphic to the multiplicative group of all r...
- 18.18.3: Fill in the blanks in the following table to obtain a group isomorp...
- 18.18.4: Repeat 18.3, with Z4 replaced by Z2 x Z2.
- 18.18.5: Assume that H = (u, v, w, x, y, z) is a group with respect to multi...
- 18.18.6: Assume that H = (u, v, w, x, y, z) is a group with respect to multi...
- 18.18.7: One of the conditions in the definition of isomorphism was not used...
- 18.18.8: Describe an isomorphism between the two groups in Example 5.5.
- 18.18.9: Prove that if G, H, and K are groups, and 8 : G -;. Hand : H -;. K ...
- 18.18.10: Prove that if G and H are groups, then G x H "" H x G. (Let the ope...
- 18.18.11: Prove that 8(x) = eX defines an isomorphism of the group lII. of al...
- 18.18.12: Verify that Z4 (operation Ell) is isomorphic to Z; (operation 0). (...
- 18.18.13: Use the mapping 8([a16) = ([ah, [ah) to show that Z6 "" Z2 X Z3. Fi...
- 18.18.14: Prove that if m and n are relatively prime (that is, have greatest ...
- 18.18.15: Assume that G, H, and 8 are as in Theorem 18.2. Assume also that B ...
- 18.18.16: The group G of all real matrices [~ ~ ], with a =f. 0, is a subgrou...
- 18.18.17: Prove that the group of rotations of a tetrahedron ( 8.18) is isomo...
Solutions for Chapter 18: ISOMORPHISM
Full solutions for Modern Algebra: An Introduction | 6th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
cond(A) = c(A) = IIAIlIIA-III = 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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.