 3.3.28E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.1E: Give a bigO estimate for the number of operations (where an operat...
 3.3.2E: Give a bigO estimate for the number additions used in this segment...
 3.3.3E: Give a bigO estimate for the number of operations, where an operat...
 3.3.4E: Give a bigO estimate for the number of operations, where an operat...
 3.3.5E: How many comparisons are used by the algorithm given in Exercise 16...
 3.3.6E: a) Use pseudocode to describe the algorithm that puts the first fou...
 3.3.7E: Suppose that an element is known to be among the first four element...
 3.3.9E: Give a bigO estimate for the number of comparisons used by the alg...
 3.3.10E: a) Show that this algorithm determines the number of 1 bits in the ...
 3.3.11E: a) Suppose we have n subsets S1. S2,…, Sn of the set {1, 2, …, n}. ...
 3.3.12E: Consider the following algorithm, which takes as input a sequence o...
 3.3.13E: The conventional algorithm for evaluating a polynomial anxn + an1x...
 3.3.14E: There is a more efficient algorithm (in terms of the number of mult...
 3.3.15E: What is the largest n for which one can solve within one second u p...
 3.3.16E: What is the largest n for which one can solve within a day using an...
 3.3.17E: What is the largest n for which one can solve within a minute using...
 3.3.18E: How much time does an algorithm take to solve a problem of size n i...
 3.3.19E: How much time does an algorithm using 250 operations need if each o...
 3.3.20E: What is the effect in the time required to solve a problem when you...
 3.3.21E: What is the effect in the time required to solve a problem when you...
 3.3.23E: Analyze the averagecase performance of the linear search algorithm...
 3.3.24E: An algorithm is called optimal for the solution of a problem with r...
 3.3.25E: Describe the worstcase time complexity, measured in terms of compa...
 3.3.26E: Describe the worstcase time complexity, measured in terms of compa...
 3.3.27E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.29E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.30E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.31E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.32E: Determine the worstcase complexity in terms of comparisons of the ...
 3.3.33E: Determine the worstcase complexity in terms of comparisons of the ...
 3.3.34E: How many comparisons does the selection sort (see preamble to Exerc...
 3.3.35E: Find a bigO estimate for the worstcase complexity in terms of num...
 3.3.36E: Show that the greedy algorithm for making change for n cents using ...
 3.3.37E: Find the complexity of a bruteforce algorithm for scheduling the t...
 3.3.38E: Find the complexity of the greedy algorithm for scheduling the most...
 3.3.39E: Describe how the number of comparisons used in the worst case chang...
 3.3.40E: Describe how the number of comparisons used in the worst case chang...
 3.3.41E: An n x n matrix is called upper triangular if aij = 0 whenever i > ...
 3.3.42E: Give a pseudocode description of the algorithm in Exercise 41 for m...
 3.3.43E: An matrix is called upper triangular if whenever .How many multipli...
 3.3.44E: What is the best order to form the product ABC if A, B, and C are m...
 3.3.45E: What is the best order to form the product ABCD if A, B, C. and D a...
 3.3.46E: In this exercise we deal with the problem of string matching.a) Exp...
Solutions for Chapter 3.3: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 3.3
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 44 problems in chapter 3.3 have been answered, more than 354511 students have viewed full stepbystep solutions from this chapter. Chapter 3.3 includes 44 full stepbystep solutions.

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.

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.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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 ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.