 4.4.1: For each of the values of n and d given in 16, find integers q and ...
 4.4.2: For each of the values of n and d given in 16, find integers q and ...
 4.4.3: For each of the values of n and d given in 16, find integers q and ...
 4.4.4: For each of the values of n and d given in 16, find integers q and ...
 4.4.5: For each of the values of n and d given in 16, find integers q and ...
 4.4.6: For each of the values of n and d given in 16, find integers q and ...
 4.4.7: Evaluate the expressions in 710
 4.4.8: Evaluate the expressions in 711
 4.4.9: Evaluate the expressions in 712
 4.4.10: Evaluate the expressions in 713
 4.4.11: Evaluate the expressions in 710
 4.4.12: Justify formula (4.4.1) for general values of DayT and N
 4.4.13: On a Monday a friend says he will meet you again in 30 days. What d...
 4.4.14: If today is Tuesday, what day of the week will it be 1,000 days fro...
 4.4.15: January 1, 2000, was a Saturday, and 2000 was a leap year. What day...
 4.4.16: Suppose d is a positive integer and n is any integer. If d  n, wha...
 4.4.17: Prove that the product of any two consecutive integers is even.
 4.4.18: The result of exercise 17 suggests that the second apparent blind a...
 4.4.19: Prove that for all integers n, n2 n + 3 is odd.
 4.4.20: Suppose a is an integer. If a mod 7 = 4, what is 5a mod 7? In other...
 4.4.21: Suppose a is an integer. If a mod 7 = 4, what is 5a mod 7? In other...
 4.4.22: Suppose a is an integer. If a mod 7 = 4, what is 5a mod 7? In other...
 4.4.23: Prove that for all integers n, if n mod 5 = 3 then n2 mod 5 = 4
 4.4.24: Prove that for all integers m and n, if m mod 5 = 2 and n mod 3 = 6...
 4.4.25: Prove that for all integers a and b, if a mod 7 = 5 and b mod 7 = 6...
 4.4.26: Prove that a necessary and sufficient condition for a nonnegative i...
 4.4.27: Prove that a necessary and sufficient condition for a nonnegative i...
 4.4.28: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.29: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.30: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.31: In 3133, you may use the properties listed in Example 4.2.3.
 4.4.32: In 3133, you may use the properties listed in Example 4.2.3.
 4.4.33: In 3133, you may use the properties listed in Example 4.2.3.
 4.4.34: Given any integer n, if n > 3, could n, n + 2, and n + 4 all be pri...
 4.4.35: Prove each of the statements in 3546
 4.4.36: Prove each of the statements in 3546
 4.4.37: Prove each of the statements in 3546
 4.4.38: Prove each of the statements in 3546
 4.4.39: Prove each of the statements in 3546
 4.4.40: Prove each of the statements in 3546
 4.4.41: Prove each of the statements in 3546
 4.4.42: Prove each of the statements in 3546
 4.4.43: Prove each of the statements in 3546
 4.4.44: Prove each of the statements in 3546
 4.4.45: Prove each of the statements in 3546
 4.4.46: Prove each of the statements in 3546
 4.4.47: A matrix M has 3 rows and 4 columns. a11 a12 a13 a14 a21 a22 a23 a2...
 4.4.48: Let M be a matrix with m rows and n columns, and suppose that the e...
 4.4.49: If m, n, and d are integers, d > 0, and m mod d = n mod d, does it ...
 4.4.50: If m, n, and d are integers, d > 0, and d (m n), what is the relat...
 4.4.51: If m, n, a, b, and d are integers, d > 0, and m mod d = a and n mod...
 4.4.52: If m, n, a, b, and d are integers, d > 0, and m mod d = a and n mod...
 4.4.53: Prove that if m, d, and k are integers and d > 0, then (m + dk) mod...
Solutions for Chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the QuotientRemainder Theorem
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the QuotientRemainder Theorem
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 53 problems in chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the QuotientRemainder Theorem have been answered, more than 48116 students have viewed full stepbystep solutions from this chapter. Chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the QuotientRemainder Theorem includes 53 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.)

Elimination.
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.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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).

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.