 6.3.1: By looking for subsets closed under the group operation, then check...
 6.3.2: In the group(a) find two different subgroups that have 3 elements.(...
 6.3.3: Prove that if G is a group and H is a subgroup of G, then the inver...
 6.3.4: Prove that if H and K are subgroups of a group G, then is a subgrou...
 6.3.5: Prove that if is a family of subgroups of a group G, thenis a subgr...
 6.3.6: Give an example of a group G and subgroups H and K of G such thatis...
 6.3.7: Let G be a group and H be a subgroup of G. (a) If G is abelian, mus...
 6.3.8: Let G be a group. If H is a subgroup of G and K is a subgroup of H,...
 6.3.9: Find the order of each element of the group
 6.3.10: List all generators of each cyclic group in Exercise 9.
 6.3.11: Let G be a group with identity e and let Prove that the setcalled t...
 6.3.12: Let G be a group and let Prove that C,the center of G, is a subgrou...
 6.3.13: Prove that if G is a group and then the center of G is a subgroup o...
 6.3.14: "Let G be a group and let H be a subgroup of G. Let a be a fixed el...
 6.3.15: Let be the group of nonzero complex numbers with complexnumber mult...
 6.3.16: Prove that for every natural number m greater than 1, the group isc...
 6.3.17: Prove that every subgroup of a cyclic group is cyclic.
 6.3.18: Let be a cyclic group of order 30.(a) What is the order of (b) List...
 6.3.19: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 6.3: Subgroups
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 6.3: Subgroups
Get Full SolutionsThis 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. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7. Since 19 problems in chapter 6.3: Subgroups have been answered, more than 5829 students have viewed full stepbystep solutions from this chapter. Chapter 6.3: Subgroups includes 19 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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