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# Solutions for Chapter 2.SE: Sequences and Summations

## Full solutions for Discrete Mathematics and Its Applications | 7th Edition

ISBN: 9780073383095

Solutions for Chapter 2.SE: Sequences and Summations

Solutions for Chapter 2.SE
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##### ISBN: 9780073383095

Summary of Chapter 2.SE: Sequences and Summations

Sequences are ordered lists of elements, used in discrete mathematics in many ways. For example, they can be used to represent solutions to certain counting problems

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.SE: Sequences and Summations includes 41 full step-by-step solutions. Since 41 problems in chapter 2.SE: Sequences and Summations have been answered, more than 698464 students have viewed full step-by-step solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Key Math Terms and definitions covered in this textbook
• 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).

• Column space C (A) =

space of all combinations of the columns of A.

• 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.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• 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).

• Hankel matrix H.

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

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• 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 .

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• 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.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Vector v in Rn.

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

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.