- 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
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.)
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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, ... , Aj-Ib. 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).
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}.