- 54.54.1: Verify that, in the proof of Theorem 54.1, e is a homomorphism of H...
- 54.54.2: Verify that, in the proof of Theorem 54.2, e is a homomorphism with...
- 54.54.3: Assume that A and B are subgroups of a group G. Prove that AB is a ...
- 54.54.4: Assume that A and B are normal subgroups of G. Then AB is a subgrou...
- 54.54.5: Assume that N <l G and that T) : G -+ G / N is the natural homomorp...
- 54.54.6: Prove: If G is a simple Abelian group, then its order is I or a prime
- 54.54.7: Prove: If G is a simple solvable group, then its order is I or a pr...
- 54.54.8: Prove that a direct product G I X G2 X ... x G" of groups is solvab...
- 54.54.9: The commutator [a, b] of elements a and b in a group (J is defined ...
- 54.54.10: Assume G is a group with commutator subgroup G' ( 54.9).(a) Prove t...
- 54.54.11: The derived series of a group G is defined by GO = G, G 1 G 2 . whe...
Solutions for Chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS
Full solutions for Modern Algebra: An Introduction | 6th Edition
Tv = Av + Vo = linear transformation plus shift.
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Every v in V is orthogonal to every w in W.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Constant down each diagonal = time-invariant (shift-invariant) filter.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.