 3.4.1: Which of these relations on the given set are antisymmetric?
 3.4.2: Let Give an example of a relation on A that is(a) antisymmetric and...
 3.4.3: Let R be an antisymmetric relation on the set A and x,(a) Prove tha...
 3.4.4: ) Give an example of a relation R on a set A that is antisymmetric ...
 3.4.5: Show that the relation R on given by for some integeris a partial o...
 3.4.6: Define the relation R on by and Provethat R is a partial ordering for
 3.4.7: Define the relation R on by Is (a + bi) R (c + di) iff a Ra partial...
 3.4.8: Let A be a partially ordered set, called the alphabet. Let W be the...
 3.4.9: Draw the Hasse diagram for the poset with the set inclusion relatio...
 3.4.10: For each Hasse diagram, list all pairs of elements in the relation ...
 3.4.11: Use your own judgment about which tasks should precede others to dr...
 3.4.12: Let A be a nonempty set and let be partially ordered by set inclusi...
 3.4.13: Let R be the rectangle shown here, including the edges. LetH be the...
 3.4.14: Let A be a set and be the ordering for (a) Let C and D be subsets o...
 3.4.15: Which are linear orders on ? Prove your answers.(a) T, where m T n ...
 3.4.16: Prove that the relation V in Exercise 15(b) is a well ordering.
 3.4.17: In determining whether a given relation is a well ordering, it is n...
 3.4.18: Prove that every subset of a wellordered set is well ordered.
 3.4.19: This exercise provides the steps necessary to prove that every part...
 3.4.20: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 3.4: Relations and Partitions
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 3.4: Relations and Partitions
Get Full SolutionsSince 20 problems in chapter 3.4: Relations and Partitions have been answered, more than 5850 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. Chapter 3.4: Relations and Partitions includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Column space C (A) =
space of all combinations of the columns of A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. 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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.