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Solutions for Chapter 1.6: Equations and Inequalities Involving Absolute Value

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 1.6: Equations and Inequalities Involving Absolute Value

Solutions for Chapter 1.6
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

This expansive textbook survival guide covers the following chapters and their solutions. Since 95 problems in chapter 1.6: Equations and Inequalities Involving Absolute Value have been answered, more than 7872 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 1.6: Equations and Inequalities Involving Absolute Value includes 95 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • 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.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Spectral Theorem A = QAQT.

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

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • 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·

  • Vector space V.

    Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

  • Wavelets Wjk(t).

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

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