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Solutions for Chapter 36: FACTORIZATION OF POLYNOMIALS

Modern Algebra: An Introduction | 6th Edition | ISBN: 9780470384435 | Authors: John R. Durbin

Full solutions for Modern Algebra: An Introduction | 6th Edition

ISBN: 9780470384435

Modern Algebra: An Introduction | 6th Edition | ISBN: 9780470384435 | Authors: John R. Durbin

Solutions for Chapter 36: FACTORIZATION OF POLYNOMIALS

Solutions for Chapter 36
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Textbook: Modern Algebra: An Introduction
Edition: 6
Author: John R. Durbin
ISBN: 9780470384435

Since 24 problems in chapter 36: FACTORIZATION OF POLYNOMIALS have been answered, more than 7899 students have viewed full step-by-step solutions from this chapter. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Chapter 36: FACTORIZATION OF POLYNOMIALS includes 24 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

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

    The n roots are the eigenvalues of A.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • 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.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • 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).

  • Row picture of Ax = b.

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

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Trace of A

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

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • 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.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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