Solutions for Chapter 21: Smallest Counterexample

Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Solutions for Chapter 21: Smallest Counterexample

This expansive textbook survival guide covers the following chapters and their solutions. Since 11 problems in chapter 21: Smallest Counterexample have been answered, more than 4300 students have viewed full step-by-step solutions from this chapter. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Chapter 21: Smallest Counterexample includes 11 full step-by-step solutions. This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3.

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.

  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • 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.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • 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.

  • 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.

  • 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.

  • Fundamental Theorem.

    The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • 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 nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Lucas numbers

    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.

  • Network.

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

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Singular Value Decomposition

    (SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

  • 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.

  • Wavelets Wjk(t).

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

×
Log in to StudySoup
Get Full Access to Thousands of Study Materials at Your School

Forgot password? Reset password here

Join StudySoup for FREE
Get Full Access to Thousands of Study Materials at Your School
Join with Email
Already have an account? Login here
Reset your password

I don't want to reset my password

Need help? Contact support

Need an Account? Is not associated with an account
Sign up
We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Got it, thanks!
Password Reset Request Sent An email has been sent to the email address associated to your account. Follow the link in the email to reset your password. If you're having trouble finding our email please check your spam folder
Got it, thanks!
Already have an Account? Is already in use
Log in
Incorrect Password The password used to log in with this account is incorrect
Try Again

Forgot password? Reset it here