 5.5.1: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.2: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.3: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.4: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.5: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.6: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.7: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.8: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.9: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.10: In 5.15.10, decide whether the given set of numbers forms a group ...
 5.5.11: Verify that {2m : m E Z} is a group with respect to multiplication....
 5.5.12: Verify that {2m 3" : m, n E Z} is a group with respect to multiplic...
 5.5.13: Let F denote M(JR), the set of all mappings from JR to JR. For f, g...
 5.5.14: Let H denote the set of all f : JR 7 JR such that f (x) 'I 0 for ...
 5.5.15: If 151> I, then M(S) is not a group with respect to composition. Why?
 5.5.16: Let G denote the set of all 2 x 2 real matrices A with det(A) 'I 0 ...
 5.5.17: Let G denote the set of all 2 x 2 real matrices with determinant eq...
 5.5.18: Verify the associative law for the operation * in Example 5.5. (Not...
 5.5.19: Consider the group in Example 5.8.(a) Verify the claim that the inv...
 5.5.20: If {a, b) with operation * is to be a group, with a the identity el...
 5.5.21: If {x, y, z} with operation * is to be a group, with x the identity...
 5.5.22: Prove: If G is a group, a E G, and a * b = b for some bEG, then a i...
 5.5.23: There are four assumptions in Theorem 5.1(a):e * a = a for each a E...
 5.5.24: Assume 5 is a nonempty set and G is a group. Let GS denote the set ...
Solutions for Chapter 5: DEFINITION AND EXAMPLES
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 5: DEFINITION AND EXAMPLES
Get Full SolutionsChapter 5: DEFINITION AND EXAMPLES includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Since 24 problems in chapter 5: DEFINITION AND EXAMPLES have been answered, more than 8966 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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!

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.