- 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 PHI-FUNCTION
- 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 TWO-DIMENSIONAL 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
Tv = Av + Vo = linear transformation plus shift.
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.
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.
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.
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, off-diagonal entries = 0.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
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 = M-I 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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