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Solutions for Chapter 9: NEW FUNCTIONS FROM OLD

Explorations in College Algebra | 5th Edition | ISBN: 9780470466445 | Authors: Linda Almgren Kime

Full solutions for Explorations in College Algebra | 5th Edition

ISBN: 9780470466445

Explorations in College Algebra | 5th Edition | ISBN: 9780470466445 | Authors: Linda Almgren Kime

Solutions for Chapter 9: NEW FUNCTIONS FROM OLD

Solutions for Chapter 9
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Textbook: Explorations in College Algebra
Edition: 5
Author: Linda Almgren Kime
ISBN: 9780470466445

This textbook survival guide was created for the textbook: Explorations in College Algebra, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 176 problems in chapter 9: NEW FUNCTIONS FROM OLD have been answered, more than 8262 students have viewed full step-by-step solutions from this chapter. Explorations in College Algebra was written by and is associated to the ISBN: 9780470466445. Chapter 9: NEW FUNCTIONS FROM OLD includes 176 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • 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.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • Diagonal matrix D.

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

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • 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.

  • Full column rank r = n.

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

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • 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.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • 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).

  • Outer product uv T

    = column times row = rank one matrix.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

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

    Column vectors by convention.

  • Similar matrices A and B.

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

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

    Spectral radius = max of IAi I.

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