 Chapter 13.1: Ballots in which voters are asked to rank all the candidates are ca...
 Chapter 13.2: If there are three or more candidates in an election, it often happ...
 Chapter 13.3: The voting method in which each candidate is compared with each of ...
 Chapter 13.4: In an election with n candidates, the number of comparisons, C, tha...
 Chapter 13.5: The voting method in which the candidate receiving the most firstp...
 Chapter 13.6: The voting method that may involve a series of elections or elimina...
 Chapter 13.7: One of the voting methods requires voters to rank all candidates fr...
 Chapter 13.8: True or False: The choice of voting method can affect an elections ...
 Chapter 13.9: In 69, the Theater Society members are voting for the kind of play ...
 Chapter 13.10: In 1013, four candidates, A, B, C, and D, are running for chair of ...
 Chapter 13.11: In 1013, four candidates, A, B, C, and D, are running for chair of ...
 Chapter 13.12: In 1013, four candidates, A, B, C, and D, are running for chair of ...
 Chapter 13.13: In 1013, four candidates, A, B, C, and D, are running for chair of ...
 Chapter 13.14: In 1416, voters in a small town are considering four proposals, A, ...
 Chapter 13.15: In 1416, voters in a small town are considering four proposals, A, ...
 Chapter 13.16: In 1416, voters in a small town are considering four proposals, A, ...
 Chapter 13.17: Use the following preference table to solve 1718. Number of Votes 1...
 Chapter 13.18: Use the following preference table to solve 1718. Number of Votes 1...
 Chapter 13.19: Use the following preference table to solve 1923. Number of Votes 1...
 Chapter 13.20: Use the following preference table to solve 1923. Number of Votes 1...
 Chapter 13.21: Use the following preference table to solve 1923. Number of Votes 1...
 Chapter 13.22: Use the following preference table to solve 1923. Number of Votes 1...
 Chapter 13.23: Use the following preference table to solve 1923. Number of Votes 1...
 Chapter 13.24: Use the following preference table, which shows the results of a st...
 Chapter 13.25: Use the following preference table, which shows the results of a st...
 Chapter 13.26: Use the following preference table to solve 2629. Number of Votes 4...
 Chapter 13.27: Use the following preference table to solve 2629. Number of Votes 4...
 Chapter 13.28: Use the following preference table to solve 2629. Number of Votes 4...
 Chapter 13.29: Use the following preference table to solve 2629. Number of Votes 4...
 Chapter 13.30: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.31: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.32: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.33: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.34: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.35: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.36: In 3036, an HMO has 40 doctors to be apportioned among four clinics...
 Chapter 13.37: In 3740, a country is composed of four states, A, B, C, and D. The ...
 Chapter 13.38: In 3740, a country is composed of four states, A, B, C, and D. The ...
 Chapter 13.39: In 3740, a country is composed of four states, A, B, C, and D. The ...
 Chapter 13.40: In 3740, a country is composed of four states, A, B, C, and D. The ...
 Chapter 13.41: A school district has 150 new laptop computers to be divided among ...
 Chapter 13.42: A country has 100 seats in the congress, divided among the three st...
 Chapter 13.43: A corporation has two branches, A and B. Each year the company awar...
 Chapter 13.44: Is the following statement true or false? There are perfect voting ...
Solutions for Chapter Chapter 13: Voting and Apportionment
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter Chapter 13: Voting and Apportionment
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Chapter Chapter 13: Voting and Apportionment includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 44 problems in chapter Chapter 13: Voting and Apportionment have been answered, more than 71931 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.