- 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 one-to-one.
- 11.1.20E: Given real-valued 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 real-valued function of a real variable. Show that if f ...
- 11.1.25E: Let f be a real-valued function of a real variable. Show that if f ...
- 11.1.26E: Let f be a real-valued 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
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].
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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).
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+ (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.
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.
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.