 6.2.1: For functions f : N R+ and g : N R+, f = O(g) if there is a positiv...
 6.2.2: For the functions f : N R+ and g : N R+ in Example 6.14 defined by ...
 6.2.3: Let f : N R+ and g : N R+ be functions defined by f(n) = 5n + 7 and...
 6.2.4: Let f(n) = 2n2 + 7n 1. Show that f = O(n3).
 6.2.5: Show that log n = O(n). [Hint: Recall that 2n n for every positive ...
 6.2.6: For which of the following is f(n) = O(n)? (a) f(n) = 100. (b) f(n)...
 6.2.7: For which of the following is f(n) = O(n2)? (a) f(n) = 2n + 5. (b) ...
 6.2.8: For the function f defined by f(n) = n2+1 n+1 for each n N, show th...
 6.2.9: Let f : N R+ and g : N R+ be two functions defined by f(n) = 2n + 1...
 6.2.10: Let f : N R+ and g : N R+ be functions defined by f(n) = n2 + 3n an...
 6.2.11: Let f : N R+ and g : N R+ be functions defined by f(n) = n2+4n+1 an...
 6.2.12: Let f : N R+ be a function. Prove that f = _(f).
 6.2.13: Prove Theorem 6.22: Let f : N R+ and g : N R+ be two functions. If ...
 6.2.14: Let f : N R+, g : N R+ and h : N R+ be three functions. Prove that ...
 6.2.15: Let f and g be two functions defined by f(n) = 1 2n2 + 5n + 1 and g...
 6.2.16: For functions f : N R+ and g : N R+, we say that f is bigomega of ...
Solutions for Chapter 6.2: Growth of Functions
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 6.2: Growth of Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.2: Growth of Functions includes 16 full stepbystep solutions. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. Since 16 problems in chapter 6.2: Growth of Functions have been answered, more than 12928 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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.

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

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 .

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.