 11.5.1E: Use the facts that to find log2(1,000), log2(1,000,000), and log2(1...
 11.5.2E: Suppose an algorithm requires operations when performed with an inp...
 11.5.5E: In 5 and 6, trace the action of the binary search algorithm (Algori...
 11.5.6E: In 5 and 6, trace the action of the binary search algorithm (Algori...
 11.5.7E: Suppose bot and top are positive integers with bot ? top. Consider ...
 11.5.8E: Exercises 8–11 refer to the following algorithm segment. For each p...
 11.5.9E: Exercises 8–11 refer to the following algorithm segment. For each p...
 11.5.10E: Exercises 8–11 refer to the following algorithm segment. For each p...
 11.5.11E: Exercises 8–11 refer to the following algorithm segment. For each p...
 11.5.12E: := n div 3Exercises 12–15 refer to the following algorithm segment....
 11.5.13E: Exercises 12–15 refer to the following algorithm segment. For each ...
 11.5.15E: Exercises 12–15 refer to the following algorithm segment. For each ...
 11.5.20E: In 20 and 21, draw a diagram like Figure 11.5.4 to show how to merg...
 11.5.21E: In 20 and 21, draw a diagram like Figure 11.5.4 to show how to merg...
 11.5.22E: In 22 and 23, draw a diagram like Figure 11.5.5 to show how merge s...
 11.5.23E: In 22 and 23, draw a diagram like Figure 11.5.5 to show how merge s...
 11.5.24E: Show that given an array a[bot], a[bot + 1], . . . , a[top] of
Solutions for Chapter 11.5: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 11.5
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Since 17 problems in chapter 11.5 have been answered, more than 27693 students have viewed full stepbystep solutions from this chapter. Chapter 11.5 includes 17 full stepbystep solutions.

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

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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