- Chapter 1: Mathematical Induction
- Chapter 10: FROM CAESAR CIPHER TO PUBLIC KEY CRYPTOGRAPHY
- Chapter 11: PERFECT NUMBERS
- Chapter 12: THE EQUATION x2 + y2 = z2
- Chapter 13: SUMS OF TWO SQUARES
- Chapter 14: THE FIBONACCI SEQUENCE
- Chapter 15: FINITE CONTINUED FRACTIONS
- Chapter 16: PRIMALITY TESTING AND FACTORIZATION
- Chapter 2: EARLY NUMBER THEORY
- Chapter 3: THE FUNDAMENTAL THEOREM OF ARITHMETIC
- Chapter 4: BASIC PROPERTIES OF CONGRUENCE
- Chapter 5: FERMAT'S LITTLE THEOREM AND PSEUDOPRIMES
- Chapter 6: THE SUM AND NUMBER OF DIVISORS
- Chapter 7: EULER'S PHI-FUNCTION
- Chapter 8: THE ORDER OF AN INTEGER MODULO n
- Chapter 9: EULER'S CRITERION
Elementary Number Theory 7th Edition - Solutions by Chapter
Full solutions for Elementary Number Theory | 7th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
Invert A by row operations on [A I] to reach [I A-I].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
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.
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).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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·
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or firstname.lastname@example.org
Forgot password? Reset it here