- 4.R.1RQ: Find 2 10 div 17 and 210 mod 17.
- 4.R.2RQ: a) Define what it means for a and b to be congruent modulo 7.______...
- 4.R.3RQ: Show that if a = b (mod m) and c = d (mod m), then a + c = b + d (m...
- 4.R.4RQ: Describe a procedure for converting decimal (base 10) expansions of...
- 4.R.5RQ: Convert (1101 1001 0101 1011)2 to octal and hexadecimal representat...
- 4.R.6RQ: Convert (7206)8 and (A0EB)16 to a binary representation.
- 4.R.7RQ: State the fundamental theorem of arithmetic.
- 4.R.8RQ: a) Describe a procedure for finding the prime factorization of an i...
- 4.R.9RQ: a) Define the greatest common divisor of two integers._____________...
- 4.R.10RQ: a) How can you find a linear combination (with integer coefficients...
- 4.R.11RQ: a) What does it mean for ? to be an inverse of a modulo m?_________...
- 4.R.12RQ: a) How can an inverse of a modulo m be used to solve the congruence...
- 4.R.13RQ: a) State the Chinese remainder theorem.________________b) Find the ...
- 4.R.14RQ: Suppose that 2n-1 = 1 (mod n). Is n necessarily prime?
- 4.R.15RQ: Use Format’s little theorem to evaluate 9200 mod 19.
- 4.R.16RQ: Explain how the check digit is found for a 10-digit ISBN.
- 4.R.17RQ: Encrypt the message APPLES AND ORANGES using a shift cipher with ke...
- 4.R.18RQ: a) What is the difference between a public key and a private key cr...
- 4.R.19RQ: Explain how encryption and decryption are done in the RSA cryptosys...
- 4.R.20RQ: Describe how two parties can share a secret key using the Diffie-He...
Solutions for Chapter 4.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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].
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.