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# Solutions for Chapter 11.7: Systems of Inequalities

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

ISBN: 9780132854351

Solutions for Chapter 11.7: Systems of Inequalities

Solutions for Chapter 11.7
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##### ISBN: 9780132854351

Chapter 11.7: Systems of Inequalities includes 61 full step-by-step solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Since 61 problems in chapter 11.7: Systems of Inequalities have been answered, more than 59690 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Distributive Law

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

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Hypercube matrix pl.

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

• 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.

• 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.

• 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.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > 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 .

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• 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 ().

• 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.

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

Spectral radius = max of IAi I.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.