 9.9.33E: In exercise 23 of Section 9.8, let Ck be the event that the gambler...
 9.9.1E: Suppose P(A  B) = 1/2 and P(A ? B) = 1/6. What is P(B)?
 9.9.2E: Suppose P(X  Y ) = 1/3 and P(Y ) = 1/4. What is P(X ? Y )?
 9.9.3E: The instructor of a discrete mathematics class gave two tests. Twen...
 9.9.4E: a. Prove that if A and B are any events in a sample space S, with P...
 9.9.5E: Suppose that A and B are events in a sample space S and that P(A), ...
 9.9.6E: An urn contains 25 red balls and 15 blue balls. Two are chosen at r...
 9.9.8E: A pool of 10 semifinalists for a job consists of 7 men and 3 women....
 9.9.9E: Prove Bayes’ Theorem for n = 2. That is, prove that if a sample spa...
 9.9.10E: Prove the full version of Bayes’ Theorem.
 9.9.11E: One urn contains 12 blue balls and 7 white balls, and a second urn ...
 9.9.12E: Redo exercise 11 assuming that the first urn contains 4 blue balls ...
 9.9.13E: One urn contains 10 red balls and 25 green balls, and a second urn ...
 9.9.14E: A drugscreening test is used in a large population of people of wh...
 9.9.15E: Two different factories both produce a certain automobile part. The...
 9.9.16E: Three different suppliers—X, Y , and Z—provide produce for a grocer...
 9.9.17E: Prove that if A and B are events in a sample space S with the prope...
 9.9.18E: Prove that if P(A ? B) = P(A) · P(B), P(A) ? 0, and P(B) ? 0, then ...
 9.9.19E: A pair of fair dice, one blue and the other gray, are rolled. Let A...
 9.9.20E: Suppose a fair coin is tossed three times. Let A be the event that ...
 9.9.21E: If A and B are events in a sample space S and A ? B = ?, what must ...
 9.9.22E: Prove that if A and B are independent events in a sample space S, t...
 9.9.23E: A student taking a multiplechoice exam does not know the answers t...
 9.9.24E: A company uses two proofreaders X and Y to check a certain manuscri...
 9.9.25E: A coin is loaded so that the probability of heads is 0.7 and the pr...
 9.9.27E: The example used to introduce conditional probability described a f...
 9.9.28E: A coin is loaded so that the probability of heads is 0.7 and the pr...
 9.9.29E: Suppose that ten items are chosen at random from a large batch deli...
 9.9.30E: Suppose the probability of a false positive result on a mammogram i...
 9.9.31E: Empirical data indicate that approximately 103 out of every 200 chi...
 9.9.32E: In exercise 23 of Section 9.8, let ?Ck ?be the event that the gambl...
Solutions for Chapter 9.9: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 9.9
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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 •

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric 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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.