 4.3.1E: Give a reason for your answer. Assume that all variables represent ...
 4.3.2E: Give a reason for your answer. Assume that all variables represent ...
 4.3.3E: Give a reason for your answer. Assume that all variables represent ...
 4.3.4E: Give a reason for your answer. Assume that all variables represent ...
 4.3.5E: Give a reason for your answer. Assume that all variables represent ...
 4.3.6E: Give a reason for your answer. Assume that all variables represent ...
 4.3.7E: Give a reason for your answer. Assume that all variables represent ...
 4.3.8E: Give a reason for your answer. Assume that all variables represent ...
 4.3.9E: Give a reason for your answer. Assume that all variables represent ...
 4.3.10E: Give a reason for your answer. Assume that all variables represent ...
 4.3.11E: Give a reason for your answer. Assume that all variables represent ...
 4.3.12E: Give a reason for your answer. Assume that all variables represent ...
 4.3.13E: Give a reason for your answer. Assume that all variables represent ...
 4.3.14E: Fill in the blanks in the following proof that for all integers a a...
 4.3.15E: Prove statement directly from the definition of divisibility.Exerci...
 4.3.16E: Prove statement directly from the definition of divisibility.Exerci...
 4.3.17E: Consider the following statement: The negative of any multiple of 3...
 4.3.18E: Show that the following statement is false: For all integers a and ...
 4.3.19E: For the statement, determine whether the statement is true or false...
 4.3.20E: For the statement, determine whether the statement is true or false...
 4.3.21E: For the statement, determine whether the statement is true or false...
 4.3.22E: For the statement, determine whether the statement is true or false...
 4.3.23E: For the statement, determine whether the statement is true or false...
 4.3.24E: For the statement, determine whether the statement is true or false...
 4.3.25E: For the statement, determine whether the statement is true or false...
 4.3.26E: For the statement, determine whether the statement is true or false...
 4.3.27E: For the statement, determine whether the statement is true or false...
 4.3.28E: For the statement, determine whether the statement is true or false...
 4.3.29E: For the statement, determine whether the statement is true or false...
 4.3.30E: For the statement, determine whether the statement is true or false...
 4.3.31E: For the statement, determine whether the statement is true or false...
 4.3.32E: A fastfood chain has a contest in which a card with numbers on it ...
 4.3.33E: Is it possible to have a combination of nickels, dimes, and quarter...
 4.3.34E: Is it possible to have 50 coins, made up of pennies, dimes, and qua...
 4.3.35E: Two athletes run a circular track at a steady pace so that the firs...
 4.3.36E: It can be shown (see exercises 14) that an integer is divisible by...
 4.3.37E: Use the unique factorization theorem to write the following integer...
 4.3.38E: Suppose that in standard factored form a = where k is a positive in...
 4.3.39E: Suppose that in standard factored form a = where k is a positive in...
 4.3.40E: a. If a and b are integers and 12a = 25b, does 12  b? does 25  a?...
 4.3.41E: How many zeros are at the end of 458 • 885? Explain how you can ans...
 4.3.42E: If n is an integer and n > 1, then n! is the product of n and every...
 4.3.43E: In a certain town of the adult men are married to of the adult wome...
 4.3.44E: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.45E: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.46E: Prove that if the decimal representation of a nonnegative integer n...
 4.3.47E: Observe that Since the sum of the digits of 7524 is divisible by 9,...
 4.3.48E: Prove that for any nonnegative integer n, if the sum of the digits ...
 4.3.49E: Given a positive integer n written in decimal form, the alternating...
Solutions for Chapter 4.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.3
Get Full SolutionsDiscrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Chapter 4.3 includes 49 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 49 problems in chapter 4.3 have been answered, more than 23966 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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 ().

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.

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

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

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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