 15.1.15.1.1: A candidate who receives more than 50% of the votes in an election ...
 15.1.15.1.2: A table that summarizes the results of an election is called a(n) t...
 15.1.15.1.3: When using the pairwise comparison method, the number of comparison...
 15.1.15.1.4: When there are four candidates, the number of comparisons needed wi...
 15.1.15.1.5: The voting method in which each voter votes for one candidate and t...
 15.1.15.1.6: The voting method in which voters rank candidates from most favorab...
 15.1.15.1.7: The voting method in which each candidate is compared with each of ...
 15.1.15.1.8: The voting method that may involve a series of elections until a ca...
 15.1.15.1.9: Plurality Three candidates are running for mayor of Tucson. They re...
 15.1.15.1.10: Plurality Method Five candidates are running for president of the S...
 15.1.15.1.11: Preference Table for Spaghetti Sauce Nine voters are asked to rank ...
 15.1.15.1.12: Preference Table for Yogurt Eight voters are asked to rank three br...
 15.1.15.1.13: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.14: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.15: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.16: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.17: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.18: Logo Choice In Exercises 1318, employees at the Bloomfield Fire Dep...
 15.1.15.1.19: Choosing a Vacation Destination In Exercises 1922, the Zellner fami...
 15.1.15.1.20: Choosing a Vacation Destination In Exercises 1922, the Zellner fami...
 15.1.15.1.21: Choosing a Vacation Destination In Exercises 1922, the Zellner fami...
 15.1.15.1.22: Choosing a Vacation Destination In Exercises 1922, the Zellner fami...
 15.1.15.1.23: NFL Expansion In Exercises 2326, the National Football League (NFL)...
 15.1.15.1.24: NFL Expansion In Exercises 2326, the National Football League (NFL)...
 15.1.15.1.25: NFL Expansion In Exercises 2326, the National Football League (NFL)...
 15.1.15.1.26: NFL Expansion In Exercises 2326, the National Football League (NFL)...
 15.1.15.1.27: Board of Trustees Election In Exercises 2731, 12 members of the exe...
 15.1.15.1.28: Board of Trustees Election In Exercises 2731, 12 members of the exe...
 15.1.15.1.29: Board of Trustees Election In Exercises 2731, 12 members of the exe...
 15.1.15.1.30: Board of Trustees Election In Exercises 2731, 12 members of the exe...
 15.1.15.1.31: Board of Trustees Election In Exercises 2731, 12 members of the exe...
 15.1.15.1.32: Post Office Sites In Exercises 3236, the 11 members of the Henriett...
 15.1.15.1.33: Post Office Sites In Exercises 3236, the 11 members of the Henriett...
 15.1.15.1.34: Post Office Sites In Exercises 3236, the 11 members of the Henriett...
 15.1.15.1.35: Post Office Sites In Exercises 3236, the 11 members of the Henriett...
 15.1.15.1.36: Post Office Sites In Exercises 3236, the 11 members of the Henriett...
 15.1.15.1.37: Choosing a Contractor The board of directors of Birds Eye Foods is ...
 15.1.15.1.38: PrimeTime Programming The programmers at the RetroVision cable tel...
 15.1.15.1.39: Flowers in a Garden The flowers in a garden at a resort need to be ...
 15.1.15.1.40: Choosing a Computer The Wizards Computer Club is ordering five new ...
 15.1.15.1.41: Describe a benefit of the Borda count method over the plurality met...
 15.1.15.1.42: Describe one way other than flipping a coin to settle a tied election.
 15.1.15.1.43: A Lost Preference Table A ski club is having an election for presid...
 15.1.15.1.44: Cat Competition Friskies Cat Food is looking for a new cat for the ...
 15.1.15.1.45: Ranking the Swim Teams The results of a swim meet between the Comet...
 15.1.15.1.46: Using the Borda Count Method Suppose that 20 voters rank three cand...
 15.1.15.1.47: Suppose that 15 voters rank four candidates. a) What is the total n...
 15.1.15.1.48: Using Approval Voting In many corporations and professional societi...
 15.1.15.1.49: Construct a preference table showing 12 votes for 3 candidates, A, ...
 15.1.15.1.50: Awards Methods Research and write a report on how voting is conduct...
Solutions for Chapter 15.1: Voting and Apportionment
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 15.1: Voting and Apportionment
Get Full SolutionsThis textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 15.1: Voting and Apportionment includes 50 full stepbystep solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 15.1: Voting and Apportionment have been answered, more than 71542 students have viewed full stepbystep solutions from this chapter.

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.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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 •

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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