×
×

# Solutions for Chapter 11.2: Discrete Mathematics and Its Applications 7th Edition

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

ISBN: 9780073383095

Solutions for Chapter 11.2

Solutions for Chapter 11.2
4 5 0 416 Reviews
18
2
##### ISBN: 9780073383095

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. This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 11.2 have been answered, more than 197946 students have viewed full step-by-step solutions from this chapter. Chapter 11.2 includes 41 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Change of basis matrix M.

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

• Cofactor Cij.

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

• Condition number

cond(A) = c(A) = IIAIlIIA-III = 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.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hankel matrix H.

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

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• 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 .

• Multiplication Ax

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

• Outer product uv T

= column times row = rank one matrix.

• Particular solution x p.

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

• 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.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Skew-symmetric matrix K.

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

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B II·

×