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Solutions for Chapter 39: Factoring

Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Solutions for Chapter 39: Factoring

Solutions for Chapter 39
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Textbook: Mathematics: A Discrete Introduction
Edition: 3
Author: Edward A. Scheinerman
ISBN: 9780840049421

This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter 39: Factoring have been answered, more than 8950 students have viewed full step-by-step solutions from this chapter. Chapter 39: Factoring includes 28 full step-by-step solutions. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421.

Key Math Terms and definitions covered in this textbook
  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • 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].

  • Full column rank r = n.

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

  • 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.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • 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).

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Orthogonal subspaces.

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

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Solvable system Ax = b.

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

  • Spectral Theorem A = QAQT.

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

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • 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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