 4.3.1: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.2: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.3: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.4: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.5: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.6: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.7: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.8: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.9: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.10: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.11: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.12: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.13: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.14: Fill in the blanks in the following proof that for all integers a a...
 4.3.15: Fill in the blanks in the following proof that for all integers a a...
 4.3.16: Fill in the blanks in the following proof that for all integers a a...
 4.3.17: Consider the following statement: The negative of any multiple of 3...
 4.3.18: Show that the following statement is false: For all integers a and ...
 4.3.19: For each statement in 1931, determine whether the statement is true...
 4.3.20: For each statement in 1931, determine whether the statement is true...
 4.3.21: For each statement in 1931, determine whether the statement is true...
 4.3.22: For each statement in 1931, determine whether the statement is true...
 4.3.23: For each statement in 1931, determine whether the statement is true...
 4.3.24: For each statement in 1931, determine whether the statement is true...
 4.3.25: For each statement in 1931, determine whether the statement is true...
 4.3.26: For each statement in 1931, determine whether the statement is true...
 4.3.27: For each statement in 1931, determine whether the statement is true...
 4.3.28: For each statement in 1931, determine whether the statement is true...
 4.3.29: For each statement in 1931, determine whether the statement is true...
 4.3.30: For each statement in 1931, determine whether the statement is true...
 4.3.31: For each statement in 1931, determine whether the statement is true...
 4.3.32: A fastfood chain has a contest in which a card with numbers on it ...
 4.3.33: Is it possible to have a combination of nickels, dimes, and quarter...
 4.3.34: Is it possible to have 50 coins, made up of pennies, dimes, and qua...
 4.3.35: Two athletes run a circular track at a steady pace so that the firs...
 4.3.36: It can be shown (see exercises 4448) that an integer is divisible b...
 4.3.37: Use the unique factorization theorem to write the following integer...
 4.3.38: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.3.39: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.3.40: a. If a and b are integers and 12a = 25b, does 12  b? does 25  a?...
 4.3.41: How many zeros are at the end of 458 885? Explain how you can answe...
 4.3.42: If n is an integer and n > 1, then n! is the product of n and every...
 4.3.43: In a certain town 2/3 of the adult men are married to 3/5 of the ad...
 4.3.44: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.45: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.46: Prove that if the decimal representation of a nonnegative integer n...
 4.3.47: Observe that 7524 = 71000 + 5100 + 210 + 4 = 7(999 + 1) + 5(99 + 1)...
 4.3.48: Prove that for any nonnegative integer n, if the sum of the digits ...
 4.3.49: Given a positive integer n written in decimal form, the alternating...
Solutions for Chapter 4.3: Direct Proof and Counterexample III: Divisibility
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.3: Direct Proof and Counterexample III: Divisibility
Get Full SolutionsDiscrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Chapter 4.3: Direct Proof and Counterexample III: Divisibility includes 49 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Since 49 problems in chapter 4.3: Direct Proof and Counterexample III: Divisibility have been answered, more than 24689 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.