 15.1E: Prove Theorem 15.1.Reference:
 15.2E: Prove Theorem 15.2.Reference:
 15.3E: Prove Theorem 15.3.Reference:
 15.4E: Prove Theorem 15.4.Reference:
 15.5E: Show that the correspondence x ? 5x from Z5 to Z10 does not preserv...
 15.6E: Show that the correspondence x ? 3x from Z4 to Z12 does not preserv...
 15.7E: Show that the mapping ?: D ? F in the proof of Theorem 15.6 is a ri...
 15.8E: Prove that every ring homomorphism f from Zn to itself has the form...
 15.9E: Suppose that ? is a ring homomorphism from Zm to Zn. Prove that if ...
 15.10E: a. Is the ring 2Z isomorphic to the ring 3Z?b. Is the ring 2Z isomo...
 15.11E: Prove that the intersection of any collection of subfields of a fie...
 15.12E: Let Z3[i] = {a+ bi  a, b ? Z3} (see Example 9 in Chapter 13). Show...
 15.13E: Let
 15.14E: Let Show that and H are isomorphic as rings.
 15.15E: Consider the mapping from M2(Z) into Z given by Prove or disprove t...
 15.16E: Let Prove or disprove that the mapping is a ring homomorphism.
 15.17E: Is the mapping from Z5 to Z30 given by x ? 6x a ring homomorphism? ...
 15.18E: Is the mapping from Z10 to Z10 given by x ? 2x a ring homomorphism?
 15.19E: Describe the kernel of the homomorphism given in Example 3.Referenc...
 15.20E: Recall that a ring element a is called an idempotent if a2 = a. Pro...
 15.21E: Determine all ring homomorphisms from Z6 to Z6. Determine all ring ...
 15.22E: Determine all ring isomorphisms from Zn to itself.
 15.23E: Determine all ring homomorphisms from Z to Z.
 15.24E: Suppose ? is a ring homomorphism from Z ? Z into Z ? Z. What are th...
 15.25E: Determine all ring homomorphisms from Z ? Z into Z ? Z.
 15.26E: In Z, let . Show that the group A/B is isomorphic to the group Z4 b...
 15.27E: Let R be a ring with unity and let ? be a ring homomorphism from R ...
 15.28E: Show that is ringisomorphic to Za ? Zb.
 15.29E: Determine all ring homomorphisms from Z ? Z to Z.
 15.30E: Prove that the sum of the squares of three consecutive integers can...
 15.31E: Let m be a positive integer and let n be an integer obtained from m...
 15.32E: (Test for Divisibility by 11) Let n be an integer with decimal repr...
 15.33E: Show that the number 7,176,825,942,116,027,211 is divisible by 9 bu...
 15.34E: Show that the number 9,897,654,527,609,805 is divisible by 99.
 15.35E: (Test for Divisibility by 3) Let n be an integer with decimal repre...
 15.36E: (Test for Divisibility by 4) Let n be an integer with decimal repre...
 15.37E: Show that no integer of the form 111,111,111, . . . ,111 is prime.
 15.38E: Consider an integer n of the form a,111,111,111,111,111,111, 111,11...
 15.39E: Suppose n is a positive integer written in the form n = ak3k + ak–1...
 15.40E: Find an analog of the condition given in the previous exercise for ...
 15.41E: In your head, determine (2 ·1075 + 2)100 mod 3 and (10100 + 1)99 mo...
 15.42E: Determine all ring homomorphisms from Q to Q.
 15.43E: Let R and S be commutative rings with unity. If f is a homomorphism...
 15.44E: Let R be a commutative ring of prime characteristic p. Show that th...
 15.45E: Is there a ring homomorphism from the reals to some ring whose kern...
 15.46E: Show that a homomorphism from a field onto a ring with more than on...
 15.47E: Suppose that R and S are commutative rings with unities. Let f be a...
 15.48E: A principal ideal ring is a ring with the property that every ideal...
 15.49E: Let R and S be rings.a. Show that the mapping from R ? S onto R giv...
 15.50E: Show that if m and n are distinct positive integers, then mZ is not...
 15.51E: Prove or disprove that the field of real numbers is ringisomorphic...
 15.52E: Show that the only ring automorphism of the real numbers is the ide...
 15.53E: Determine all ring homomorphisms from R to R.
 15.54E: Suppose that n divides m and that a is an idempotent of Zn (that is...
 15.55E: Show that the operation of multiplication defined in the proof of T...
 15.56E: Let a, b ? Q}. Show that these two rings are not ringisomorphic.
 15.57E: Let Z[i] = {a + bi  a, b ? Z}. Show that the field of quotients of...
 15.58E: Let F be a field. Show that the field of quotients of F is ringisom...
 15.59E: Let D be an integral domain and let F be the field of quotients of ...
 15.60E: Let D be an integral domain and let F be the field of quotients of ...
 15.61E: Show that the relation ?defined in the proof of Theorem 15.6 is an ...
 15.62E: Give an example of a ring without unity that is contained in a field.
 15.63E: Prove that the set T in the proof of Corollary 3 to Theorem 15.5 is...
 15.64E: Suppose that ?: R ? S is a ring homomorphism and that the image of ...
 15.65E: Let ? (x) ? R[x]. If a + bi is a complex zero of ? (x) (here ), sho...
 15.66E: a. Show that ? is a homomorphism.b. Determine the kernel of ?.c. Sh...
 15.67E: Show that the prime subfield of a field of characteristic p is ring...
 15.68E: Let n be a positive integer. Show that there is a ring isomorphism ...
 15.69E: Show that Zmn is ringisomorphic to Zm ? Zn when m and n are relati...
 15.70E: Prove that every integer with decimal representation of the form ab...
Solutions for Chapter 15: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 15
Get Full SolutionsChapter 15 includes 70 full stepbystep solutions. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. This expansive textbook survival guide covers the following chapters and their solutions. Since 70 problems in chapter 15 have been answered, more than 15364 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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