 2.1CE: This software produces the following information about U(n) (see Ex...
 2.1E: Which of the following binary operations are closed?a. subtraction ...
 2.2CE: This software produces the following information about U(n) (see Ex...
 2.2E: Which of the following binary operations are associative?a. multipl...
 2.3CE: This software produces the following information about U(n) (see Ex...
 2.3E: Which of the following binary operations are commutative?a. substra...
 2.4CE: This software produces the following information about U(n) (see Ex...
 2.4E: Which of the following sets are closed under the given operation?a....
 2.5E: In each case, find the inverse of the element under the given opera...
 2.6E: In each case, perform the indicated operation.
 2.7E: Give two reasons why the set of odd integers under addition is not ...
 2.8E: Referring to Example 13, verify the assertion that subtraction is n...
 2.9E: Show that {1, 2, 3} under multiplication modulo 4 is not a group bu...
 2.10E: Show that the group GL(2, R) of Example 9 is nonAbelian by exhibit...
 2.11E: Find the inverse of the element
 2.12E: Give an example of group elements a and b with the property that a–...
 2.13E: Translate each of the following multiplicative expressions into its...
 2.14E: For group elements a, b, and c, express (ab)3 and (ab–2 c)2 withou...
 2.15E: Let G be a group and let Show that G = H as sets.
 2.16E: Show that the set {5, 15, 25, 35} is a group under multiplication m...
 2.17E: (From the GRE Practice Exam)* Let p and q be distinct primes. Suppo...
 2.18E: List the members of
 2.19E: Prove that the set of all 2 × 2 matrices with entries from R and de...
 2.20E: For any integer n > 2, show that there are at least two elements in...
 2.21E: An abstract algebra teacher intended to give a typist a list of nin...
 2.22E: Let G be a group with the property that for any x, y, z in the grou...
 2.23E: (Law of Exponents for Abelian Groups) Let a and b be elements of an...
 2.24E: (Socks–Shoes Property) Draw an analogy between the statement (ab)–1...
 2.25E: Prove that a group G is Abelian if and only if (ab)–1 = a–1b–1 for ...
 2.26E: Prove that in a group, (a–1)–1 = a for all a.
 2.27E: For any elements a and b from a group and any integer n, prove that...
 2.28E: If a1, a2, . . . , an belong to a group, what is the inverse of a1a...
 2.29E: The integers 5 and 15 are among a collection of 12 integers that fo...
 2.30E: Give an example of a group with 105 elements. Give two examples of ...
 2.31E: Prove that every group table is a Latin square†; that is, each elem...
 2.32E: Construct a Cayley table for U(12).
 2.33E: Suppose the table below is a group table. Fill in the blank entries.
 2.34E: Prove that in a group, (ab)2 = a2b2 if and only if ab = ba.
 2.35E: Let a, b, and c be elements of a group. Solve the equation axb = c ...
 2.36E: Let a and b belong to a group G. Find an x in G such that xabx–1 = ba.
 2.37E: Let G be a finite group. Show that the number of elements x of G su...
 2.38E: Give an example of a group with elements a, b, c, d, and x such tha...
 2.39E: Suppose that G is a group with the property that for every choice o...
 2.40E: Find an element X in D4 such that R90VXH = D'.
 2.41E: Suppose F1 and F2 are distinct reflections in a dihedral group Dn. ...
 2.42E: Suppose F1 and F2 are distinct reflections in a dihedral group Dn s...
 2.43E: Let R be any fixed rotation and F any fixed reflection in a dihedra...
 2.44E: Let R be any fixed rotation and F any fixed reflection in a dihedra...
 2.45E: In the dihedral group Dn, let R = R360/n and let F be any reflectio...
 2.46E: Prove that the set of all rational numbers of the form 3m6n, where ...
 2.47E: Prove that if G is a group with the property that the square of eve...
 2.48E: Prove that the set of all 3 × 3 matrices with real entries of the f...
 2.49E: Prove the assertion made in Example 20 that the set {1, 2, . . . , ...
 2.50E: In a finite group, show that the number of nonidentity elements tha...
 2.51E: List the six elements of GL(2, Z2). Show that this group is non Ab...
 2.52E: Let . Show that G is a group under matrix multiplication. Explain w...
 2.53E: Suppose that in the definition of a group G, the condition that the...
 2.54E: Suppose that in the definition of a group G, the condition that for...
Solutions for Chapter 2: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 2
Get Full SolutionsChapter 2 includes 58 full stepbystep solutions. Since 58 problems in chapter 2 have been answered, more than 15348 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th.

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.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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