 11.3.1: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.2: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.3: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.4: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.5: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.6: For Exercises 16, find the maximum and minimum values of thequadrat...
 11.3.7: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.8: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.9: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.10: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.11: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.12: For Exercises 712, find the maximum and minimum values ofthe quadra...
 11.3.13: For Exercises 1316, find the maximum and minimum values ofthe quadr...
 11.3.14: For Exercises 1316, find the maximum and minimum values ofthe quadr...
 11.3.15: For Exercises 1316, find the maximum and minimum values ofthe quadr...
 11.3.16: For Exercises 1316, find the maximum and minimum values ofthe quadr...
 11.3.17: For Exercises 1720, find the maximum and minimum values ofthe quadr...
 11.3.18: For Exercises 1720, find the maximum and minimum values ofthe quadr...
 11.3.19: For Exercises 1720, find the maximum and minimum values ofthe quadr...
 11.3.20: For Exercises 1720, find the maximum and minimum values ofthe quadr...
 11.3.21: For Exercises 2124, find the maximum and minimum values ofthe quadr...
 11.3.22: For Exercises 2124, find the maximum and minimum values ofthe quadr...
 11.3.23: For Exercises 2124, find the maximum and minimum values ofthe quadr...
 11.3.24: For Exercises 2124, find the maximum and minimum values ofthe quadr...
 11.3.25: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.26: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.27: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.28: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.29: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.30: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 11.3.31: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.32: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.33: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.34: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.35: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.36: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 11.3.37: Prove Theorem 11.11: Suppose that Q : Rn Ris a quadraticformQ(x) = ...
 11.3.38: Prove the following generalized version of Theorem 11.12: LetA be a...
 11.3.39: Prove that if Q(x) is a quadratic form and c is a constant, thenQ(c...
Solutions for Chapter 11.3: Constrained Optimization
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 11.3: Constrained Optimization
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. Chapter 11.3: Constrained Optimization includes 39 full stepbystep solutions. Since 39 problems in chapter 11.3: Constrained Optimization have been answered, more than 16955 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.)

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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 IIA1 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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.