 Chapter 1: MAPPINGS
 Chapter 10: CONGRUENCE. THE DIVISION ALGORITHM
 Chapter 11: INTEGERS MODULO n
 Chapter 12: GREATEST COMMON DIVISORS. THE EUCLIDEAN ALGORITHM
 Chapter 13: FACTORIZATION. EULER'S PHIFUNCTION
 Chapter 14: ELEMENTARY PROPERTIES
 Chapter 15: GENERATORS. DIRECT PRODUCTS
 Chapter 16: COSETS
 Chapter 17: LAGRANGE'S THEOREM. CYCLIC GROUPS
 Chapter 18: ISOMORPHISM
 Chapter 19: MORE ON ISOMORPHISM
 Chapter 2: COMPOSITION. INVERTIBLE MAPPINGS
 Chapter 20: CAYLEY'S THEOREM
 Chapter 21: HOMOMORPHISMS OF GROUPS. KERNELS
 Chapter 22: QUOTIENT GROUPS
 Chapter 23: THE FUNDAMENTAL HOMOMORPHISM THEOREM
 Chapter 24: DEFINITION AND EXAMPLES
 Chapter 25: INTEGRAL DOMAINS. SUBRINGS
 Chapter 26: FIELDS
 Chapter 27: ISOMORPHISM. CHARACTERISTIC
 Chapter 28: ORDERED INTEGRAL DOMAINS
 Chapter 29: THE INTEGERS
 Chapter 3: OPERATIONS
 Chapter 30: FIELD OF QUOTIENTS. THE FIELD OF RATIONAL NUMBERS
 Chapter 31: ORDERED FIELDS. THE FIELD OF REAL NUMBERS
 Chapter 32: THE FIELD OF COMPLEX NUMBERS
 Chapter 33: COMPLEX ROOTS OF UNITY
 Chapter 34: DEFINITION AND ELEMENTARY PROPERTIES
 Chapter 35: THE DIVISION ALGORITHM
 Chapter 36: FACTORIZATION OF POLYNOMIALS
 Chapter 37: UNIQUE FACTORIZATION DOMAINS
 Chapter 38: HOMOMORPHISMS OF RINGS. IDEALS
 Chapter 39: QUOTIENT RINGS
 Chapter 4: COMPOSITION AS AN OPERATION
 Chapter 40: QUOTIENT RINGS OF F[X]
 Chapter 41: FACTORIZATION AND IDEALS
 Chapter 42: SIMPLE EXTENSIONS. DEGREE
 Chapter 43: ROOTS OF POLYNOMIALS
 Chapter 44: FUNDAMENTAL THEOREM: INTRODUCTION
 Chapter 45: ALGEBRAIC EXTENSIONS
 Chapter 46: SPLITTING FIELDS. GALOIS GROUPS
 Chapter 47: SEPARABILITY AND NORMALITY
 Chapter 48: FUNDAMENTAL THEOREM OF GALOIS THEORY
 Chapter 49: SOLVABILITY BY RADICALS
 Chapter 5: DEFINITION AND EXAMPLES
 Chapter 50: FINITE FIELDS
 Chapter 51: THREE FAMOUS PROBLEMS
 Chapter 52: CONSTRUCTIBLE NUMBERS
 Chapter 53: IMPOSSIBLE CONSTRUCTIONS
 Chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS
 Chapter 55: ALTERNATING GROUPS
 Chapter 56: GROUPS ACTING ON SETS
 Chapter 57: BURNSIDE'S COUNTING THEOREM
 Chapter 58: SYLOW'S THEOREM
 Chapter 59: FINITE SYMMETRY GROUPS
 Chapter 6: PERMUTATIONS
 Chapter 60: INFINITE TWODIMENSIONAL SYMMETRY GROUPS
 Chapter 61: ON CRYSTALLOGRAPHIC GROUPS
 Chapter 62: THE EUCLIDEAN GROUP
 Chapter 63: PARTIALLY ORDERED SETS
 Chapter 64: LATTICES
 Chapter 65: BOOLEAN ALGEBRAS
 Chapter 66: FINITE BOOLEAN ALGEBRAS
 Chapter 7: SUBGROUPS
 Chapter 8: GROUPS AND SYMMETRY
 Chapter 9: EQUIVALENCE RELATIONS
Modern Algebra: An Introduction 6th Edition  Solutions by Chapter
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Modern Algebra: An Introduction  6th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Modern Algebra: An Introduction were answered by Patricia, our top Math solution expert on 03/16/18, 02:52PM. This expansive textbook survival guide covers the following chapters: 66. Modern Algebra: An Introduction was written by Patricia and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since problems from 66 chapters in Modern Algebra: An Introduction have been answered, more than 4076 students have viewed full stepbystep answer.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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·
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