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Solutions for Chapter A.2: Discrete Mathematics and Its Applications 7th Edition

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Full solutions for Discrete Mathematics and Its Applications | 7th Edition

ISBN: 9780073383095

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Solutions for Chapter A.2

Since 6 problems in chapter A.2 have been answered, more than 196496 students have viewed full step-by-step solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions. Chapter A.2 includes 6 full step-by-step solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • 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.

  • 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

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • 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.

  • Orthogonal subspaces.

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

  • Outer product uv T

    = column times row = rank one matrix.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Rank r (A)

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

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Skew-symmetric matrix K.

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

  • Spectral Theorem A = QAQT.

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

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

  • Trace of A

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

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Vector v in Rn.

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

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