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Textbooks / Math / Discrete Mathematics with Applications 4

Discrete Mathematics with Applications 4th Edition - Solutions by Chapter

Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp

Full solutions for Discrete Mathematics with Applications | 4th Edition

ISBN: 9780495391326

Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp

Discrete Mathematics with Applications | 4th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 278 Reviews
Textbook: Discrete Mathematics with Applications
Edition: 4
Author: Susanna S. Epp
ISBN: 9780495391326

This expansive textbook survival guide covers the following chapters: 131. Since problems from 131 chapters in Discrete Mathematics with Applications have been answered, more than 34234 students have viewed full step-by-step answer. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. The full step-by-step solution to problem in Discrete Mathematics with Applications were answered by , our top Math solution expert on 07/19/17, 06:34AM. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Key Math Terms and definitions covered in this textbook
  • Complete solution x = x p + Xn to Ax = b.

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

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • 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].

  • 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

  • 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.

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

    Use AT for complex A.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • 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.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • 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.

  • 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.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • 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).

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • 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.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vector v in Rn.

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

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