 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 2n1 = 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 10digit 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 DiffieHe...
Solutions for Chapter 4.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 4.R
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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.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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.