 4.2.16E: Determine which of the statements in 15–20 are true and which are f...
 4.2.24E: Derive the statements in 24–26 as corollaries of Theorems 4.2.1, 4....
 4.2.25E: Derive the statements in 24–26 as corollaries of Theorems 4.2.1, 4....
 4.2.26E: Derive the statements in 24–26 as corollaries of Theorems 4.2.1, 4....
 4.2.34E: Prove the statement. In each case use only the definitions of the t...
 4.2.1E: The number are all rational. Write each number as a ratio of two in...
 4.2.2E: The number are all rational. Write each number as a ratio of two in...
 4.2.3E: The number are all rational. Write each number as a ratio of two in...
 4.2.4E: The number are all rational. Write each number as a ratio of two in...
 4.2.5E: The number are all rational. Write each number as a ratio of two in...
 4.2.6E: The number are all rational. Write each number as a ratio of two in...
 4.2.7E: The number are all rational. Write each number as a ratio of two in...
 4.2.8E: The zero product property, says that if a product of two real numbe...
 4.2.9E: Assume that a and b are both integers and that a ? 0 and b ? 0. Exp...
 4.2.10E: Assume that m and n are both integers and that n ? 0. Explain why (...
 4.2.11E: Prove that every integer is a rational number.
 4.2.12E: Fill in the blanks in the following proof that the square of any ra...
 4.2.13E: Consider the statement: The negative of any rational number is rati...
 4.2.14E: Consider the statement: The cube of any rational number is a ration...
 4.2.15E: Determine which of the statement are true and which are false. Prov...
 4.2.17E: Determine if the statement is true and which are false. Prove each ...
 4.2.18E: Determine which of the statement are true and which are false. Prov...
 4.2.19E: Determine which of the statement are true and which are false. Prov...
 4.2.20E: Determine which of the statement are true and which are false. Prov...
 4.2.21E: Use the properties of even and odd integers that are listed in Exam...
 4.2.22E: Use the properties of even and odd integers that are listed in Exam...
 4.2.23E: Use the properties of even and odd integers that are listed in Exam...
 4.2.27E: It is a fact that if n is any nonnegative integer, then.(A more gen...
 4.2.28E: Suppose a, b, c, and d are integers and a ? c. Suppose also that x ...
 4.2.29E: Suppose a, b, and c are integers and x, y, and z are nonzero real n...
 4.2.30E: Prove that if one solution for a quadratic equation of the form x2 ...
 4.2.31E: Prove that if a real number c satisfies a polynomial equation of th...
 4.2.32E: Prove that for all real numbers c, if c is a root of a polynomial w...
 4.2.33E: Use the properties of even and odd integers that are listed in Exam...
 4.2.35E: Find the mistakes in the “proofs” that the sum of any two rational ...
 4.2.36E: Find the mistakes in the “proofs” that the sum of any two rational ...
 4.2.37E: Find the mistakes in the “proofs” that the sum of any two rational ...
 4.2.38E: Find the mistakes in the “proofs” that the sum of any two rational ...
 4.2.39E: Find the mistakes in the “proofs” that the sum of any two rational ...
Solutions for Chapter 4.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.2
Get Full SolutionsSince 39 problems in chapter 4.2 have been answered, more than 47954 students have viewed full stepbystep solutions from this chapter. 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: 4. Chapter 4.2 includes 39 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

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

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.