 9.9.1: Solve the equation n D 720 for n
 9.9.2: There are six different French books, eight different Russian books...
 9.9.3: Give an AliceandBob discussion about what it means to add (and to...
 9.9.4: Consider the formula .n/k D n .n k/: This formula is mostly correct...
 9.9.5: Evaluate 100 98 without calculating 100 or 98.
 9.9.6: Order the following integers from least to greatest: 2 100 , 1002 ,...
 9.9.7: The Scottish mathematician James Stirling found an approximation fo...
 9.9.8: Calculate the following products: a. Q4 kD1 .2k C 1/. b. Q4 kD3 k. ...
 9.9.9: Please calculate the following: a. 1 1. b. 1 1 C 2 2. c. 1 1 C 2 2 ...
 9.9.10: When 100 is written out in full, it equals 100 D 9332621 : : : 0000...
 9.9.11: Prove that all of the following numbers are composite: 1000C2, 1000...
 9.9.12: A factorion is a positive integer with the following cute property....
 9.9.13: Can factorial be extended to negative integers? On the basis of equ...
 9.9.14: Evaluate: 0 0
 9.9.15: The double factorial n is defined for odd positive integers n; it i...
 9.9.16: Let n be a positive integer. What is the n th derivative of x n ?
 9.9.17: The following formula appears in W.A. Granvilles Elements of the Di...
 9.9.18: Evaluate the following integral for n D 0; 1; 2; 3; 4: Z 1 0 x n e ...
Solutions for Chapter 9: Factorial
Full solutions for Mathematics: A Discrete Introduction  3rd Edition
ISBN: 9780840049421
Solutions for Chapter 9: Factorial
Get Full SolutionsMathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Since 18 problems in chapter 9: Factorial have been answered, more than 9270 students have viewed full stepbystep solutions from this chapter. 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. Chapter 9: Factorial includes 18 full stepbystep solutions.

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!

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

lAII = 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.

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.

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).

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

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.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.