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Solutions for Chapter 3.3: Polynomial and Rational Functions

Algebra and Trigonometry: Real Mathematics, Real People | 7th Edition | ISBN: 9781305071735 | Authors: Ron Larson

Full solutions for Algebra and Trigonometry: Real Mathematics, Real People | 7th Edition

ISBN: 9781305071735

Algebra and Trigonometry: Real Mathematics, Real People | 7th Edition | ISBN: 9781305071735 | Authors: Ron Larson

Solutions for Chapter 3.3: Polynomial and Rational Functions

Solutions for Chapter 3.3
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Textbook: Algebra and Trigonometry: Real Mathematics, Real People
Edition: 7
Author: Ron Larson
ISBN: 9781305071735

Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Chapter 3.3: Polynomial and Rational Functions includes 120 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 120 problems in chapter 3.3: Polynomial and Rational Functions have been answered, more than 58997 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7.

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

    Tv = Av + Vo = linear transformation plus shift.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • 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.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Eigenvalue A and eigenvector x.

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

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Indefinite matrix.

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

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • 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).

  • Nullspace N (A)

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

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Orthogonal subspaces.

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

  • 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).

  • Rank r (A)

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

  • 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.

  • Trace of A

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

  • Wavelets Wjk(t).

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

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