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Solutions for Chapter 6-5: Solving Open Sentences Involving Absolute Value

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition

ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Solutions for Chapter 6-5: Solving Open Sentences Involving Absolute Value

Solutions for Chapter 6-5
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Textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1)
Edition: 1
Author: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more
ISBN: 9780078738227

This expansive textbook survival guide covers the following chapters and their solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 6-5: Solving Open Sentences Involving Absolute Value includes 70 full step-by-step solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Since 70 problems in chapter 6-5: Solving Open Sentences Involving Absolute Value have been answered, more than 37155 students have viewed full step-by-step solutions from this chapter.

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.

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • 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.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • 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.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • 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.

  • 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 •

  • Partial pivoting.

    In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Spectral Theorem A = QAQT.

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

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • 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.

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