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Solutions for Chapter 9.2: Inverse Functions

Intermediate Algebra | 6th Edition | ISBN: 9780321785046 | Authors: Elayn El Martin-Gay

Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Intermediate Algebra | 6th Edition | ISBN: 9780321785046 | Authors: Elayn El Martin-Gay

Solutions for Chapter 9.2: Inverse Functions

Solutions for Chapter 9.2
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Textbook: Intermediate Algebra
Edition: 6
Author: Elayn El Martin-Gay
ISBN: 9780321785046

Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Since 70 problems in chapter 9.2: Inverse Functions have been answered, more than 63016 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.2: Inverse Functions includes 70 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • 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).

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • 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 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.

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • 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.

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Partial pivoting.

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

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • 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).

  • Semidefinite matrix A.

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

  • Similar matrices A and B.

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

  • Spectral Theorem A = QAQT.

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

  • 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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