 7.4.1: For the following integers a, b and n, determine whether a is congr...
 7.4.2: Classify each of the following statements as true or false. (a) 24 ...
 7.4.3: Give an example of four integers that are congruent to 4 modulo 13.
 7.4.4: Determine whether each of the following integers is congruent to 5 ...
 7.4.5: Recall Theorem 7.25: Let a, b and n 2 be integers. Then a b (mod n)...
 7.4.6: Let a, b, c and n 2 be integers. Prove that if a b (mod n) and b c ...
 7.4.7: Let a, b, m and n be integers with m, n 2 and m  n. Show that if a...
 7.4.8: Let a, b, c, d and n 2 be integers. Show that if a b (mod n) and c ...
 7.4.9: Let a, b, c, d and n 2 be integers. Show that if a b (mod n) and c ...
 7.4.10: Let a, b, c, d and n 2 be integers. Show that if a b (mod n) and c ...
 7.4.11: Let a, b and n 2 be integers. Show that if a b (mod n), then ar br ...
 7.4.12: Let a, b and n 2 be integers. (a) Prove that if a b (mod n), then a...
 7.4.13: Let a, b Z. Prove that if a 0 (mod 5) and b 2 (mod 5), then a2 + b2...
 7.4.14: Let a Z. Prove that if a2 6 a (mod 3), then a 6 0 (mod 3) and a 6 1...
 7.4.15: Let a, b Z. Prove that if a2 + 2b2 0 (mod 3), then either both a an...
 7.4.16: Let a, b, c Z. Prove that if abc 1 (mod 3), then an odd number of a...
 7.4.17: For each of the following pairs a, b of integers and an integer n 2...
 7.4.18: Prove or disprove: There exist distinct integers a, b, c 2 such tha...
Solutions for Chapter 7.4: Congruence
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 7.4: Congruence
Get Full SolutionsChapter 7.4: Congruence includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. Since 18 problems in chapter 7.4: Congruence have been answered, more than 12263 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.