 5.5.1E: Exercises 1–5 contain a while loop and a predicate. In each case sh...
 5.5.2E: Exercises 1–5 contain a while loop and a predicate. In each case sh...
 5.5.3E: Exercises 1–5 contain a while loop and a predicate. In each case sh...
 5.5.4E: Exercises 1–5 contain a while loop and a predicate. In each case sh...
 5.5.5E: Exercises 1–5 contain a while loop and a predicate. In each case sh...
 5.5.6E: Exercises 6–9 each contain a while loop annotated with a preand a p...
 5.5.7E: Exercises 6–9 each contain a while loop annotated with a preand a p...
 5.5.8E: Exercises 6–9 each contain a while loop annotated with a preand a p...
 5.5.9E: Exercises 6–9 each contain a while loop annotated with a preand a p...
 5.5.10E: Prove correctness of the while loop of Algorithm 4.8.3 (in exercise...
 5.5.11E: The following while loop implements a way to multiply two numbers t...
Solutions for Chapter 5.5: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.5
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 5.5 includes 11 full stepbystep solutions. Since 11 problems in chapter 5.5 have been answered, more than 43598 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.

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!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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 ().

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.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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