 29.1E: Determine the number of ways in which the four corners of a square ...
 29.2E: Determine the number of different necklaces that can be made using ...
 29.3E: Determine the number of ways in which the vertices of an equilatera...
 29.4E: A benzene molecule can be modeled as six carbon atoms arranged in a...
 29.5E: Suppose that in Exercise 4 we permit only NH2 and COOH for the radi...
 29.6E: Determine the number of ways in which the faces of a regular dodeca...
 29.7E: Determine the number of ways in which the edges of a square can be ...
 29.8E: Determine the number of ways in which the edges of a square can be ...
 29.9E: Determine the number of different 11bead necklaces that can be mad...
 29.10E: Determine the number of ways in which the faces of a cube can be co...
 29.11E: Suppose a cake is cut into six identical pieces. How many ways can ...
 29.12E: How many ways can the five points of a fivepointed crown be painte...
 29.13E: Let G be a finite group and let sym(G) be the group of all permutat...
 29.14E: Let G be a finite group, let H be a subgroup of G, and let S be the...
 29.15E: For a fixed square, let L1 be the perpendicular bisector of the top...
Solutions for Chapter 29: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 29
Get Full SolutionsContemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. Since 15 problems in chapter 29 have been answered, more than 15367 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 29 includes 15 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
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).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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