 8.5.1E: How many elements are in A1? A2 if there are 12 elements in A1, 18 ...
 8.5.3E: A survey of households in the United States reveals that 96% have a...
 8.5.7E: There are 2504 computer science students at a school. Of these, 187...
 8.5.4E: A marketing report concerning personal computers states that 650,00...
 8.5.5E: Find the number of elements in A1 ? A2 ? A3 if there are 100 elemen...
 8.5.9E: How many students are enrolled in a course either in calculus. disc...
 8.5.8E: In a survey of 270 college students, it is found that 64 like bruss...
 8.5.11E: Find the number of positive integers not exceeding 100 that are eit...
 8.5.13E: How many bit strings of length eight do not contain six consecutive...
 8.5.15E: How many permutations of the 10 digits either begin with the 3 digi...
 8.5.16E: How many elements are in the union of four sets if each of the sets...
 8.5.17E: How many elements are in the union of four sets if the sets have 50...
 8.5.18E: How many terms are there in the formula for the number of elements ...
 8.5.19E: Write out the explicit formula given by the principle of inclusion...
 8.5.20E: How many elements are in the union of five sets if the sets contain...
 8.5.21E: Write out the explicit formula given by the principle of inclusion...
 8.5.22E: Prove the principle of inclusionexclusion using mathematical induc...
 8.5.23E: Let E1, E2 , and E3 be three events from a sample space S. Find a f...
 8.5.24E: Find the probability that when a fair coin is flipped five times ta...
 8.5.25E: Find the probability that when four numbers from 1 to 100, inclusiv...
 8.5.26E: Find a formula for the probability of the union of four events in a...
 8.5.27E: Find a formula for the probability of the union of five events in a...
 8.5.29E: Find a formula for the probability of the union of n events in a sa...
 8.5.28E: Find a formula for the probability of the union of n events in a sa...
Solutions for Chapter 8.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 8.5
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Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

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).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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 .

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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 II·