- 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 inclusion-exclusion 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
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
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 edge-node 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.
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·