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Solutions for Chapter 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: Polynomial and Rational Functions

Solutions for Chapter 3
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Textbook: Algebra and Trigonometry: Real Mathematics, Real People
Edition: 7
Author: Ron Larson
ISBN: 9781305071735

Since 125 problems in chapter 3: Polynomial and Rational Functions have been answered, more than 61741 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. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3: Polynomial and Rational Functions includes 125 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

    Use AT for complex A.

  • 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.

  • Gauss-Jordan method.

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

  • Hermitian matrix A H = AT = A.

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

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( 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.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Semidefinite matrix A.

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

  • Singular matrix A.

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

  • Singular Value Decomposition

    (SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

  • Skew-symmetric matrix K.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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