- 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
A = CTC = (L.J]))(L.J]))T for positive definite A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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.
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Solvable system Ax = b.
The right side b is in the column space of A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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