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Solutions for Chapter 12.4: Probability

A Survey of Mathematics with Applications | 9th Edition | ISBN:  9780321759665 | Authors: Allen R. Angel, Christine D. Abbott, Dennis C. Runde

Full solutions for A Survey of Mathematics with Applications | 9th Edition

ISBN: 9780321759665

A Survey of Mathematics with Applications | 9th Edition | ISBN:  9780321759665 | Authors: Allen R. Angel, Christine D. Abbott, Dennis C. Runde

Solutions for Chapter 12.4: Probability

Solutions for Chapter 12.4
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Textbook: A Survey of Mathematics with Applications
Edition: 9
Author: Allen R. Angel, Christine D. Abbott, Dennis C. Runde
ISBN: 9780321759665

Chapter 12.4: Probability includes 59 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 59 problems in chapter 12.4: Probability have been answered, more than 70856 students have viewed full step-by-step solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • 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.

  • 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].

  • 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.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Hermitian matrix A H = AT = A.

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

  • Identity matrix I (or In).

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

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Orthogonal subspaces.

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

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Right inverse A+.

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

  • Standard basis for Rn.

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

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Subspace S of V.

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

  • 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.

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