 15.4.15.1.142: When group A loses an item or items to group B even though group As...
 15.4.15.1.143: When the addition of a new group and additional items to be apporti...
 15.4.15.1.144: When an increase in the total number of items to be apportioned res...
 15.4.15.1.145: Hamiltons method and Jeffersons method favor states.
 15.4.15.1.146: Adamss method and Websters method favor states.
 15.4.15.1.147: The apportionment method that satisfies the quota rule but may prod...
 15.4.15.1.148: In Exercises 718, round quotas to the nearest hundredthFax Machines...
 15.4.15.1.149: In Exercises 718, round quotas to the nearest hundredthErgonomic Ch...
 15.4.15.1.150: In Exercises 718, round quotas to the nearest hundredthLegislative ...
 15.4.15.1.151: In Exercises 718, round quotas to the nearest hundredthLegislative ...
 15.4.15.1.152: In Exercises 1114, assume that the number of items to be apportione...
 15.4.15.1.153: In Exercises 1114, assume that the number of items to be apportione...
 15.4.15.1.154: In Exercises 1114, assume that the number of items to be apportione...
 15.4.15.1.155: In Exercises 1114, assume that the number of items to be apportione...
 15.4.15.1.156: Additional Employees Cynergy Telecommunications has employees in Eu...
 15.4.15.1.157: Adding a Park The town of Manlius purchased 25 new picnic tables to...
 15.4.15.1.158: Adding a State A country with three states has 60 seats in the legi...
 15.4.15.1.159: Adding a State A country with two states has 33 seats in the legisl...
 15.4.15.1.160: Write a paper on which apportionment method you think is the best. ...
Solutions for Chapter 15.4: Voting and Apportionment
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 15.4: Voting and Apportionment
Get Full SolutionsSince 19 problems in chapter 15.4: Voting and Apportionment have been answered, more than 74082 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 15.4: Voting and Apportionment includes 19 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665.

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

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.

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.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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

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