 8.5.6E: In Exercises 5–8, find the minimal representation of the polytope d...
 8.5.7E: In Exercises 5–8, find the minimal representation of the polytope d...
 8.5.8E: In Exercises 5–8, find the minimal representation of the polytope d...
 8.5.10E: Find an example of a closed convex set S in R2 such that its profil...
 8.5.9E: Is the origin an extreme point of conv S? Is the origin a vertex of...
 8.5.1E:
 8.5.2E:
 8.5.3E: Repeat Exercise 1 where m is the minimum value of f on S instead of...
 8.5.4E: Repeat Exercise 2 where m is the minimum value of f on S instead of...
 8.5.5E: In Exercises 5–8, find the minimal representation of the polytope d...
 8.5.11E: Find an example of a bounded convex set S in R2 such that its profi...
 8.5.12E: a. Determine the number of kfaces of the 5dimensional simplex S5 ...
 8.5.13E:
 8.5.14E:
 8.5.15E:
 8.5.16E: In Exercises 16 and 17, mark each statement True or False. Justify ...
 8.5.17E: In Exercises 16 and 17, mark each statement True or False. Justify ...
 8.5.18E: Let v be an element of the convex set S. Prove that v is an extreme...
 8.5.19E:
 8.5.20E: Find an example to show that the convexity of S is necessary in Exe...
 8.5.21E: If A and B are convex sets, prove that A + B is convex.
 8.5.22E: A polyhedron (3polytope) is called regular if all its facets are c...
Solutions for Chapter 8.5: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 8.5
Get Full SolutionsSince 22 problems in chapter 8.5 have been answered, more than 46727 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 8.5 includes 22 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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).

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

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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 .

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

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

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 space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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