 8.1: Maximize f = 2x + 3y subject to 2x + 4y :5 16 3x + 2y :5 12 x O,y 0
 8.2: Maximize f = 6x + 4 y subject to x + 2y :5 16 3x + 2y :5 24 x O,y 0
 8.3: Minimize f = 4x + y subject to 3x + 2y :5 21 x + Sy ::s 20 x O,y:::; 0
 8.4: A farmer has to decide how many acres of a 40acre plot are to be d...
 8.5: A company is buying lockers. It has narrowed the choice down to two...
 8.6: Use the simplex method to maximize f = 2x + y + z under the constra...
 8.7: A furniture company finishes two kinds of tables, X and Y. There ar...
 8.8: Minimize f = x  2y + 4z subject to x  y + 3z ::s 4 2x + 2y  3z :...
Solutions for Chapter 8: Linear Programming
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 8: Linear Programming
Get Full SolutionsSince 8 problems in chapter 8: Linear Programming have been answered, more than 9538 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8: Linear Programming includes 8 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Conjugate Gradient Method.
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. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

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

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).