- 3.3.1.: Please determine which of the following are true and which are fals...
- 3.3.2.: Here is a possible alternative to Definition 3.2: We say that a is ...
- 3.3.3: None of the following numbers is prime. Explain why they fail to sa...
- 3.3.4: The natural numbers are the nonnegative integers; that is, numbers....
- 3.3.5: A rational number is a number formed by dividing two integers a=b w...
- 3.3.6: Define what it means for an integer to be a perfect square. For exa...
- 3.3.7: Define what it means for one number to be the square root of anothe...
- 3.3.8: Define the perimeter of a polygon.
- 3.3.9: Suppose the concept of distance between points in the plane is alre...
- 3.3.10: Define the midpoint of a line segment.
- 3.3.11: Some English words are difficult to define with mathematical precis...
- 3.3.12: Discrete mathematicians especially enjoy counting problems: problem...
- 3.3.13: An integer n is called perfect provided it equals the sum of all it...
- 3.3.14: At a Little League game there are three umpires. One is an engineer...
Solutions for Chapter 3: Definition
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Upper triangular systems are solved in reverse order Xn to Xl.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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 .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
= Xl (column 1) + ... + xn(column n) = combination of columns.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Pseudoinverse A+ (Moore-Penrose 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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.