×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide

Solutions for Chapter 7: EULER'S PHI-FUNCTION

Full solutions for Elementary Number Theory | 7th Edition

ISBN: 9780073383149

Solutions for Chapter 7: EULER'S PHI-FUNCTION

Solutions for Chapter 7
4 5 0 256 Reviews
17
0
Textbook: Elementary Number Theory
Edition: 7
Author: Professor David Burton
ISBN: 9780073383149

Chapter 7: EULER'S PHI-FUNCTION includes 21 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since 21 problems in chapter 7: EULER'S PHI-FUNCTION have been answered, more than 5979 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Covariance matrix:E.

    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.

  • 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.

  • 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.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Left inverse A+.

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

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • 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.

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • 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.

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • 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 p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password