- 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 11-bead 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 five-pointed 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
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Gram-Schmidt 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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Outer product uv T
= column times row = rank one matrix.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.