×
×

# Solutions for Chapter A.10: nth Roots; Rational Exponents

## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Solutions for Chapter A.10: nth Roots; Rational Exponents

Solutions for Chapter A.10
4 5 0 264 Reviews
15
5
##### ISBN: 9780132854351

This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 116 problems in chapter A.10: nth Roots; Rational Exponents have been answered, more than 55721 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter A.10: nth Roots; Rational Exponents includes 116 full step-by-step solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Complex conjugate

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

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.

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• 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.

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Outer product uv T

= column times row = rank one matrix.

• 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.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Skew-symmetric matrix K.

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

• 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·

• Volume of box.

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

×