 11.1.1E: The graph of a function f is shown below.a. Is f (0) positive or ne...
 11.1.2E: The graph of a function g is shown below.a. Is g(0) positive or neg...
 11.1.3E: Draw the graphs of the power functions p1/3 and p1/4 on the same se...
 11.1.4E: Draw the graphs of the power functions p3 and p4 on the same set of...
 11.1.5E: Draw the graphs of for all real numbers x. What can you conclude fr...
 11.1.6E: Graph each of the functions defined in 6–9 below.g(x) = [x] for all...
 11.1.7E: Graph each of the functions defined in 6–9 below.h(x) = [x]  [x] f...
 11.1.8E: Graph each of the functions defined in 6–9 below.F(x) = [x ½] for a...
 11.1.9E: Graph each of the functions defined in 6–9 below.G(x) = x  [x] for...
 11.1.10E: In each of 10–13 a function is defined on a set of integers. Graph ...
 11.1.11E: In each of 10–13 a function is defined on a set of integers. Graph ...
 11.1.12E: In each of 10–13 a function is defined on a set of integers. Graph ...
 11.1.13E: In each of 10–13 a function is defined on a set of integers. Graph ...
 11.1.14E: The graph of a function f is shown below. Find the intervals on whi...
 11.1.15E: Show that the function f : R ? R defined by the formula f (x) = 2x ...
 11.1.16E: Show that the function g: R ? R defined by the formula g(x) = ?(x/3...
 11.1.17E: Let h be the function from R to R defined by the formula h(x) = x2 ...
 11.1.18E: Let k: R ? R be the function defined by the formula k(x) = (x ? 1)/...
 11.1.19E: Show that if a function f : R ? R is increasing, then f is onetoone.
 11.1.20E: Given realvalued functions f and g with the same domain D, the sum...
 11.1.21E: a. Let m be any positive integer, and define f (x) = xm for all non...
 11.1.22E: Let f be the function whose graph is shown below. Draw the graph of...
 11.1.23E: Let h be the function whose graph is shown below. Draw the graph of...
 11.1.24E: Let f be a realvalued function of a real variable. Show that if f ...
 11.1.25E: Let f be a realvalued function of a real variable. Show that if f ...
 11.1.26E: Let f be a realvalued function of a real variable. Show that if f ...
 11.1.28E: In 27 and 28, functions f and g are defined. In each case draw the ...
Solutions for Chapter 11.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 11.1
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 27 problems in chapter 11.1 have been answered, more than 23941 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Chapter 11.1 includes 27 full stepbystep solutions.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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).

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose 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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Solvable system Ax = b.
The right side b is in the column space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI 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.
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