- 2.R.1RQ: Explain what it means for one set to be a subset of another set. Ho...
- 2.R.2RQ: What is the empty set ? Show that the empty set is a subset of ever...
- 2.R.3RQ: a) Define | S |, the cardinality of the set S.________________b) Gi...
- 2.R.4RQ: a) Define the power set of a set S.________________b) When is the e...
- 2.R.5RQ: a) Define the union, intersection, difference, and symmetric differ...
- 2.R.6RQ: a) Explain what it means for two sets to be equal.________________b...
- 2.R.7RQ: Explain the relationship between logical equivalences and set ident...
- 2.R.8RQ: a) Define the domain, codomain, and range of a function.___________...
- 2.R.9RQ: a) Define what it means for a function from the set of positive int...
- 2.R.10RQ: a) Define the inverse of a function.________________b) When docs a ...
- 2.R.11RQ: a) Define the floor and ceiling functions from the set of real numb...
- 2.R.12RQ: Conjecture a formula for the terms of the sequence that begins 8, 1...
- 2.R.13RQ: Suppose that an = an?1 ?5 for n = 1, 2, .... Find a formula for an.
- 2.R.14RQ: What is the sum of the terms of the geometric progression a + ar +....
- 2.R.15RQ: Show that the set of odd integers is countable.
- 2.R.16RQ: Give an example of an uncountable set.
- 2.R.17RQ: Define the product of two matrices A and B. When is this product de...
- 2.R.18RQ: Show that matrix multiplication is not commutative.
Solutions for Chapter 2.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
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 n-l - An).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.