- 64.64.1: Show that the lattice on the right in Figure 64.1 is nondistributive
- 64.64.2: Construct the diagram for the lattice of subgroups of ((1 2), (3 4)...
- 64.64.3: Prove: If a partially ordered set is a chain, then it is a distribu...
- 64.64.4: Show that the lattice of divisors of 12 (Example 63.6) is not compl...
- 64.64.5: Is the lattice of subgroups of 2':6 distributive? Is it complemente...
- 64.64.6: Is the lattice of subgroups of S3 distributive? Is it complemented?...
- 64.64.7: Construct the diagram for the lattice of subgroups of the group of ...
- 64.64.8: Prove that if n E N, then the set of all positive divisors of n is ...
- 64.64.9: If S is a set, what is the complement of an element in the lattice ...
- 64.64.10: Construct the diagram for the lattice of subgroups of the group of ...
- 64.64.11: Prove the first half of Theorem 64.1.
- 64.64.12: Supply the proof that a 1\ b = a implies a v b = b, which was omitt...
- 64.64.13: The lattice N, with a :5. b defined to mean a I b, is distributive....
- 64.64.14: Explain why each distributive lattice satisfies the law a v (b 1\ c...
- 64.64.15: Lattices are isomorphic iff they are isomorphic as partially ordere...
- 64.64.16: Prove that each finite subset of a lattice has a g.l.b. and a l.u.b.
Solutions for Chapter 64: LATTICES
Full solutions for Modern Algebra: An Introduction | 6th Edition
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
A sequence of steps intended to approach the desired solution.
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.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.