 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
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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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).