 8.2.1: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.2: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.3: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.4: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.5: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.6: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.7: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.8: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.9: In Exercises 19 use the simplex method to maximize the functions u...
 8.2.10: A company uses three machines, I, II, and III to produce items X, Y...
 8.2.11: An industrial furniture company manufactures desks, cabinets, and c...
 8.2.12: A company produces washing machines at three factories, A, B, and C...
 8.2.13: A manufacturer makes three lines of tents, all from the same materi...
 8.2.14: In Exercises 1416 use the simplex method to minimize the functions...
 8.2.15: In Exercises 1416 use the simplex method to minimize the functions...
 8.2.16: In Exercises 1416 use the simplex method to minimize the functions...
Solutions for Chapter 8.2: The Simplex Method
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 8.2: The Simplex Method
Get Full SolutionsChapter 8.2: The Simplex Method includes 16 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 8.2: The Simplex Method have been answered, more than 8660 students have viewed full stepbystep solutions from this chapter.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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 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!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.