- 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
Tv = Av + Vo = linear transformation plus shift.
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.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Column space C (A) =
space of all combinations of the columns of A.
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
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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 .
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Outer product uv T
= column times row = rank one matrix.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
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).
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).
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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.
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.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.