 8.8.1.1: Sketch the feasible set with constraints x + 2y 6, 2x + y 6, x 0, y...
 8.8.2.1: Minimize x1 +x2 x3, subject to 2x1 4x2 +x3 +x4 = 4 3x1 +5x2 +x3 +x5...
 8.8.3.1: What is the dual of the following problem: Minimize x1 + x2, subjec...
 8.8.4.1: In Figure 8.5, add 3 to every capacity. Find by inspection the maxi...
 8.8.5.1: How will the optimal strategies in the game that opens this section...
 8.8.1.2: (Recommended) On the preceding feasible set, what is the minimum va...
 8.8.2.2: After the preceding simplex step, prepare for and decide on the nex...
 8.8.3.2: What is the dual of the following problem: Maximize y2 subject to y...
 8.8.4.2: Find a maximal flow and minimal cut for the following network:
 8.8.5.2: With payoff matrix A = 1 2 3 4 , explain the calculation by X of th...
 8.8.1.3: Show that the feasible set constrained by 2x+5y 3, 3x+8y 5, x 0, y ...
 8.8.2.3: In Example 3, suppose the cost is 3x + y. With rearrangement, the c...
 8.8.3.3: Suppose A is the identity matrix (so that m = n), and the vectors b...
 8.8.4.3: If you could increase the capacity of any one pipe in the network a...
 8.8.5.3: If ai j is the largest entry in its row and the smallest in its col...
 8.8.1.4: Show that the following problem is feasible but unbounded, so it ha...
 8.8.2.4: Suppose the cost function in Example 3 is x y, so that after rearra...
 8.8.3.4: Construct a 1 by 1 example in which Ax b, x 0 is unfeasible, and th...
 8.8.4.4: Draw a 5node network with capacity i j between node i and node j...
 8.8.5.4: Compute Ys best strategy by weighting the rows of A = 3 4 1 2 0 3 w...
 8.8.1.5: Add a single inequality constraint to x 0, y 0 such that the feasib...
 8.8.2.5: Again in Example 3, change the cost to x+3y. Verify that the simple...
 8.8.3.5: Starting with the 2 by 2 matrix A = 1 0 0 1 , choose b and c so tha...
 8.8.4.5: In a graph, the maximum number of paths from s to t with no common ...
 8.8.5.5: With the same A as in 4, find the best strategy for X. Show that X ...
 8.8.1.6: What shape is the feasible set x 0, y 0, z 0, x + y + z = 1, and wh...
 8.8.2.6: Phase I finds a basic feasible solution to Ax = b (a corner). After...
 8.8.3.6: If all entries of A, b, and c are positive, show that both the prim...
 8.8.4.6: Find a maximal set of marriages (a complete matching, if possible) ...
 8.8.5.6: Find both optimal strategies, and the value, if A = " 1 0 1 2 1 2 # .
 8.8.1.7: Solve the portfolio problem at the end of the preceding section.
 8.8.2.7: If we wanted to maximize instead of minimize the cost (with Ax = b ...
 8.8.3.7: Show that x = (1,1,1,0) and y = (1,1,0,1) are feasible in the prima...
 8.8.4.7: For the matrix A in 6, which rows violate Halls conditionby having ...
 8.8.5.7: Suppose A = a b c d . What weights x1 and 1 x1 will give a column o...
 8.8.1.8: In the feasible set for the General Motors problem, the nonnegativi...
 8.8.2.8: Minimize 2x1 +x2, subject to x1 +x2 4, x1 +3x2 12, x1 x2 0, x 0
 8.8.3.8: Verify that the vectors in the previous exercise satisfy the comple...
 8.8.4.8: How many lines (horizontal and vertical) are needed to cover all th...
 8.8.5.8: Find x , y and the value v for A = 1 0 0 0 2 0 0 0 3
 8.8.1.9: (Transportation problem) Suppose Texas, California, and Alaska each...
 8.8.2.9: Verify the inverse in equation (5), and show that BE has Bv = u in ...
 8.8.3.9: Suppose that A = 1 0 0 1 , b = 1 1 , and c = 1 1 . Find the optimal...
 8.8.4.9: (a) Suppose every row and every column contains exactly two 1s. Pro...
 8.8.5.9: Compute min yi0 y1+y2=1 max x10 x1+x2=1 (x1y1 +x2y2).
 8.8.2.10: Suppose we want to minimize cx = x1 x2, subject to 2x1 4x2 +x3 = 6 ...
 8.8.3.10: If the primal problem is constrained by equations instead of inequa...
 8.8.4.10: If a 7 by 7 matrix has 15 1s, prove that it allows at least 3 marri...
 8.8.5.10: Explain each of the inequalities in equation (5). Then, once the mi...
 8.8.2.11: For the matrix P = I A T (AAT ) 1A, show that if x is in the nullsp...
 8.8.3.11: (a) Without the simplex method, minimize the cost 5x1 +3x2 +4x3, su...
 8.8.4.11: For infinite sets, a complete matching may be impossible even if Ha...
 8.8.5.11: Show that x = (1 2 , 1 2 ,0,0) and y = (1 2 , 1 2 ) are optimal str...
 8.8.2.12: (a) Minimize the cost c T x = 5x1 +4x2 +8x3 on the plane x1 +x2 +x3...
 8.8.3.12: If the primal has a unique optimal solution x , and then c is chang...
 8.8.4.12: If Figure 8.5 shows lengths instead of capacities, find the shortes...
 8.8.5.12: Has it been proved that no chess strategy always wins for black? Th...
 8.8.3.13: Write the dual of the following problem: Maximize x1+x2+x3 subject ...
 8.8.4.13: Apply algorithms 1 and 2 to find a shortest spanning tree for the n...
 8.8.5.13: If X chooses a prime number and simultaneously Y guesses whether it...
 8.8.3.14: If A = 1 1 0 1 , describe the cone of nonnegative combinations of t...
 8.8.4.14: (a) Why does the greedy algorithm work for the spanning tree proble...
 8.8.5.14: If X is a quarterback, with the choice of run or pass, and Y can de...
 8.8.3.15: In three dimensions, can you find a set of six vectors whose cone o...
 8.8.4.15: If A is the 5 by 5 matrix with is just above and just below the mai...
 8.8.3.16: Use 8H to show that the following equation has no solution, because...
 8.8.4.16: The maximal flow problem has slack variables wi j = ci j xi j for t...
 8.8.3.17: Use 8I to show that there is no solution x 0 (the alternative holds...
 8.8.3.18: Show that the alternatives in 8J (Ax b, x 0, yA 0, yb < 0, y 0) can...
Solutions for Chapter 8: Linear Programming and Game Theory
Full solutions for Linear Algebra and Its Applications,  4th Edition
ISBN: 9780030105678
Solutions for Chapter 8: Linear Programming and Game Theory
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications,, edition: 4. Chapter 8: Linear Programming and Game Theory includes 69 full stepbystep solutions. 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: 9780030105678. Since 69 problems in chapter 8: Linear Programming and Game Theory have been answered, more than 10408 students have viewed full stepbystep solutions from this chapter.

Amplitude
See Sinusoid.

Cotangent
The function y = cot x

Demand curve
p = g(x), where x represents demand and p represents price

equation of a quadratic function
ƒ(x) = ax 2 + bx + c(a ? 0)

Fundamental
Theorem of Algebra A polynomial function of degree has n complex zeros (counting multiplicity).

Graphical model
A visible representation of a numerical or algebraic model.

Infinite limit
A special case of a limit that does not exist.

Intercept
Point where a curve crosses the x, y, or zaxis in a graph.

Length of a vector
See Magnitude of a vector.

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

Nonsingular matrix
A square matrix with nonzero determinant

Octants
The eight regions of space determined by the coordinate planes.

Oddeven identity
For a basic trigonometric function f, an identity relating f(x) to f(x).

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Polar coordinate system
A coordinate system whose ordered pair is based on the directed distance from a central point (the pole) and the angle measured from a ray from the pole (the polar axis)

Proportional
See Power function

Semiminor axis
The distance from the center of an ellipse to a point on the ellipse along a line perpendicular to the major axis.

Slopeintercept form (of a line)
y = mx + b

xaxis
Usually the horizontal coordinate line in a Cartesian coordinate system with positive direction to the right,.

xintercept
A point that lies on both the graph and the xaxis,.