- 13.13.1: Determine the standard/arm (13.1)/01' each a/the/allowing integers....
- 13.13.2: Determine the standard/arm (13.1)/01' each a/the/allowing integers....
- 13.13.3: Determine the standard/arm (13.1)/01' each a/the/allowing integers....
- 13.13.4: Determine the standard/arm (13.1)/01' each a/the/allowing integers....
- 13.13.5: LetImageandImagewhere PI, P2, ... ,Pk are distinct prime numbers, S...
- 13.13.6: Let m and n be as in 13.5, and letUI = the minimum of Sl and II for...
- 13.13.7: Use Ihe results of 13.6 10 compute the greatest common divisor and ...
- 13.13.8: Use Ihe results of 13.6 10 compute the greatest common divisor and ...
- 13.13.9: Use Ihe results of 13.6 10 compute the greatest common divisor and ...
- 13.13.10: Use Ihe results of 13.6 10 compute the greatest common divisor and ...
- 13.13.11: Determine all positive integral divisors of each of the following i...
- 13.13.12: Determine the number of positive integral divisors of an integer n ...
- 13.13.13: Construct the Cayley table for 1[J 12.
- 13.13.14: Find the inverse of  in 1[Jso. (Suggestion: Look at the proof o...
- 13.13.15: Prove that if n is odd, then ,p(2n) = ,p(n).
- 13.13.16: Prove that if n is even, then ,p(2n) = 2,p(n).
- 13.13.17: Prove that if (a, b) = I, a 1m, and b 1m, then ab 1m. (Suggestion: ...
- 13.13.18: An integer is square-free if it is not divisible by the square of a...
- 13.13.19: Prove that if n is a positive integer, then .Jii is rational iff n ...
- 13.13.20: Prove that --Y2 is irrational. (Suggestion: Apply the Fundamental T...
- 13.13.21: State and prove a theorem characterizing those integers n for which...
- 13.13.22: Prove that if a and b are positive integers, then(a, b)[a, b] = abo...
- 13.13.23: Prove Theorem 13.4. (Suggestion: Use Theorems 13.1 and 13.3 and mat...
Solutions for Chapter 13: FACTORIZATION. EULER'S PHI-FUNCTION
Full solutions for Modern Algebra: An Introduction | 6th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of 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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Gram-Schmidt 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.
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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).
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Solvable system Ax = b.
The right side b is in the column space of A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.