 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 43738 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 11.1 includes 27 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

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].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

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+ (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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.