 Chapter 13.1: Write down all the subsets of the set 5Dave, Joanne, Erica6.
 Chapter 13.2: If n1A2 = 8, n1B2 = 12, and n1A B2 = 3, find n1A B2.
 Chapter 13.3: If n1A2 = 12, n1A B2 = 30, and n1A B2 = 6, find n1B2.
 Chapter 13.4: How many are in A?
 Chapter 13.5: How many are in A or B?
 Chapter 13.6: How many are in A and C?
 Chapter 13.7: How many are not in B?
 Chapter 13.8: How many are in neither A nor C?
 Chapter 13.9: How many are in B but not in C?
 Chapter 13.10: In 10 and 11, compute the given expression.P18, 32
 Chapter 13.11: In 10 and 11, compute the given expression.C18, 32
 Chapter 13.12: A clothing store sells pure wool and polyesterwool suits. Each sui...
 Chapter 13.13: On a given day, the American Baseball League schedules 7 games. How...
 Chapter 13.14: If 4 people enter a bus having 9 vacant seats, in how many ways can...
 Chapter 13.15: In how many ways can a squad of 4 relay runners be chosen from a tr...
 Chapter 13.16: In how many ways can 2 teams from 14 teams in the American League b...
 Chapter 13.17: Using the digits 0, 1, 2, p , 9, how many 7digit numbers can be fo...
 Chapter 13.18: A license plate consists of 1 letter, excluding O and I, followed b...
 Chapter 13.19: Using the digits 0 and 1, how many different numbers consisting of ...
 Chapter 13.20: How many different vertical arrangements are there of 10 flags if 4...
 Chapter 13.21: A group of 9 people is going to be formed into committees of 4, 3, ...
 Chapter 13.22: For this problem, assume that a year has 365 days. (a) How many way...
 Chapter 13.23: According to the U.S. Bureau of Labor Statistics, 8.8% of the U.S. ...
 Chapter 13.24: You have four $1 bills, three $5 bills, and two $10 bills in your w...
 Chapter 13.25: Each of the numbers, 1, 2, p , 100 is written on an index card and ...
 Chapter 13.26: At the Milex tuneup and brake repair shop, the manager has found t...
 Chapter 13.34: A brick staircase has a total of 25 steps. The bottom step requires...
Solutions for Chapter Chapter 13: Sequences; Induction; the Binomial Theorem
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter Chapter 13: Sequences; Induction; the Binomial Theorem
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Chapter Chapter 13: Sequences; Induction; the Binomial Theorem includes 27 full stepbystep solutions. Since 27 problems in chapter Chapter 13: Sequences; Induction; the Binomial Theorem have been answered, more than 55975 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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