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# Solutions for Chapter 5.4: Counting

## Full solutions for Discrete Mathematics and Its Applications | 6th Edition

ISBN: 9780073229720

Solutions for Chapter 5.4: Counting

Solutions for Chapter 5.4
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##### ISBN: 9780073229720

This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Since 39 problems in chapter 5.4: Counting have been answered, more than 40939 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. Chapter 5.4: Counting includes 39 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Cramer's Rule for Ax = b.

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

• Distributive Law

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

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Outer product uv T

= column times row = rank one matrix.

• Semidefinite matrix A.

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

• 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!

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Spanning set.

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

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• 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}.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

• Volume of box.

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