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mathematics
linear algebra and its applications
Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions
In mark each statement. Justify each answer.(T/F) An orthogonal matrix is orthogonally diagonalizable.
In Exercises 1 and 2, set up the initial simplex tableau for the given linear programming problem. Maximize subject to 22x1 + 14x2 3x15x2 2x17x2 6x1 + X₂ and x₁ ≥ 0, x₂ ≥ 0. 30 24 42
For each simplex tableau in Exercises 3– 6, do the following:a. Determine which variable should be brought into the solution.b. Compute the next tableau.c. Identify the basic feasible solution corresponding to the tableau in part (b).d. Determine if the answer in part (c) is optimal. X1 X2 5
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If x is a vector whose entries sum to 1, then x is a probability vector.
For each simplex tableau in Exercises 3– 6, do the following:a. Determine which variable should be brought into the solution.b. Compute the next tableau.c. Identify the basic feasible solution corresponding to the tableau in part (b).d. Determine if the answer in part (c) is optimal. X1 X2 -1 1 1
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If x is a pure strategy in a matrix game, then all the coordinates in x have the same value.
For each simplex tableau in Exercises 3– 6, do the following:a. Determine which variable should be brought into the solution.b. Compute the next tableau.c. Identify the basic feasible solution corresponding to the tableau in part (b).d. Determine if the answer in part (c) is optimal.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.Each strategy for player R in a matrix game is a convex combination of the set of pure strategies for R.
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.A slack variable is used to change an equality into an inequality.
Solve Exercises 13–18 by using the simplex method. Minimize subject to 2x1 + 3x₂ + 3x3 x1 - 2x₂ -8 15 x3 ≤ 25 2x2 + x3 x₂ + x2 2x1 - and x₁ ≥ 0, x₂ > 0, X3 ≥ 0.
Solve Exercises 13–18 by using the simplex method. Minimize subject to 12x1 + 5x₂ 2x1 + x₂ ≥ 32 -3x1 + 5x₂ ≤ 30 and x₁0, x₂ > 0.
Solve Exercises 13–18 by using the simplex method. Maximize subject to 2x1 + 5x2 + 3x3 x1 + 2x₂ 2x1 < 28 +4x316 x₂ + x3 ≤ 12 and x1 ≥ 0, x2 ≥ 0, X3 ≥ 0.
Solve Exercises 13–18 by using the simplex method. Maximize subject to 4x1 + 5x2 x1 + 2x₂ < 26 2x13x230 x1 + x₂ ≤ 13 VI and x₁ ≥ 0, x₂ ≥ 0.
Solve Exercises 13–18 by using the simplex method. Maximize subject to 5x1 + 4x₂ x1 + 5x₂ ≤ 70 3x1 + 2x2 ≤ 54 and x₁ ≥ 0, x₂ ≥ 0.
Solve Exercises 13–18 by using the simplex method. Maximize subject to 10x₁ + 12x2 2x13x236 5x1 + 4x2 55 and x₁ ≥ 0, x₂ > 0.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f (x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.If the primal problem has an optimal solution, then the final tableau in the simplex method
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f(x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.If x̅ is an optimal solution to the primal problem and ŷ is a feasible solution to the dual
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If a canonical linear programming problem is unbounded, then it must be feasible.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f(x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.If either the primal or the dual problem has an optimal solution, then they both do.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If x is an extreme point of the feasible set of a canonical linear programming problem, then x is an optimal solution.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If the objective function in a canonical linear programming problem takes on arbitrarily large values in the feasible set, then the problem is infeasible.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f(x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.If both the primal and the dual problems are feasible, then they both have optimal solutions.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f (x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.The dual of the dual problem is the original primal problem.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If A is the payoff matrix for a 2 x n matrix game, then the value of strategy x(t) to player R, denoted ν(x(t)), is the maximum value of n linear functions of t.
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.The bottom entry in the right column of a simplex tableau gives the maximum value of the objective
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.The basic feasible solutions correspond to the extreme points of the feasible set.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If a canonical linear programming problem has a feasible solution but no optimal solution, then the objective function must be unbounded on the feasible set.
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.If one of the coordinates in vector b is negative, then the problem is infeasible.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If A is the payoff matrix for a matrix game, then the value of strategy x to player R, denoted ν(x), is the minimum of the inner product of x with each of the columns of A.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f(x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.The dual problem is to minimize y in Rm subject to Ay ≥ c and y ≥ 0.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If x̂ and ŷ are optimal strategies for an m x n matrix game whose value is v, then ŷ is a convex combination of the pure strategies ej in Rn for which E(x̂,ej) = ν.
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.A solution is feasible if each variable is nonnegative.
Exercises 7–12 relate to a canonical linear programming problem with an m x n coefficient matrix A in the constraint inequality Ax ≤ b. Mark each statement True or False (T/F). Justify each answer.A solution is called a basic solution if m or fewer of the variables are nonzero.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If A is an m x n payoff matrix, then the strategy space for R is the set of all probability vectors in Rn.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.A strategy x̂ for row player R is optimal if the value of x̂ is equal to the value of the game to R.
For each simplex tableau in Exercises 3– 6, do the following:a. Determine which variable should be brought into the solution.b. Compute the next tableau.c. Identify the basic feasible solution corresponding to the tableau in part (b).d. Determine if the answer in part (c) is optimal.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If the feasible set of a canonical linear programming problem is unbounded, then the program has no optimal solution.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f (x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each answer.If a slack variable is in an optimal solution, then the marginal value of the item
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.The simplex method of solving a canonical linear programming problem begins by changing each constraint inequality into an equality.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If A is m x n, then it will require n slack variables to change Ax ≤ b into a system of linear equations.
In Exercises 17–20, use the simplex method to solve the dual, and from this solve the original problem (the dual of the dual). Minimize subject to 16x₁ + 10x2 + 20x3 x₁ + X2 + 2x₁ + x₂ + and x₁ ≥ 0, x₂ ≥ 0, X3 ≥ 0. X2 3x3 ≥ 4 2x3 ≥ 5
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.In following the simplex method, when a variable goes “out” of a basic feasible solution, it stays out.
Use the simplex method to solve the linear programming problem in Section 9.2, Example 1.Data From Section 9.2 Example 1 EXAMPLE 1 The Shady-Lane grass seed company blends two types of seed mix- tures, EverGreen and QuickGreen. Each bag of EverGreen contains 3 pounds of fescue seed, 1 pound of rye
The bottom row of the final tableau for a linear programming problem will have zeros as entries in the columns corresponding to the basic variables that are “in” the solution. There may also be zeros in some of the other columns in this bottom row. When this happens, the optimal solution will
Try to use the simplex method to solve the problem in Exercise 27. Explain why it doesn’t work.Data From Exercise 27Consider the following problem: Maximize x₁ + x₂ subject to -x₁ + x₂ ≤1 X2 ≤2 and X₁ ≥ 0, x₂ ≥ 0.
Solve the matrix game in Exercise 10 in Section 9.1 using linear programming.Data From Section 9.1 Exercise 10 Let M be the 2 0 1-2 matrix game having payoff matrix 1 1 -2 2 1 x and y have the given values. -1 1 -1 . Find E(x, y), u(x), and v(y) when
Use the simplex method to find an optimal solution to the problem in Exercise 25.Data From Exercise 25Consider the following problem: Maximize subject to -X₁ + 2x₂ -x₁ + and x₁ ≥ 0, x2 ≥ 0. x₂ ≤1 X2 ≤2
Consider the matrix game having payoff matrix Find the optimal mixed strategies and the value of the game by using the method of Example 4 in Section 9.1.Data From Section 9.1 Example 4 A = [ -1 2 4 3 3 -2 ]
Bob wishes to invest $35,000 in stocks, bonds, and gold coins. He knows that his rate of return will depend on the economic climate of the country, which is, of course, difficult to predict. After careful analysis, he determines the annual profit in dollars he would expect per hundred dollars on
Solve the matrix games in Exercises 23 and 24 by using linear programming. 2 -4 -1 0 5 3
Use the simplex method to solve the linear programming problem in Section 9.2, Exercise 1.Data From Section 9.2 Exercise 1 Betty plans to invest a total of $12,000 in mutual funds, cer- tificates of deposit (CDs), and a high-yield savings account. Because of the risk involved in mutual funds, she
Use the simplex method to solve the linear programming problem in Section 9.2, Exercise 17.Data From Section 9.2 Exercise 17 The Benri Company manufactures two kinds of kitchen gad- gets: invertible widgets and collapsible whammies. The pro- duction process is divided into three departments:
In Exercises 17–20, use the simplex method to solve the dual, and from this solve the original problem (the dual of the dual). Minimize subject to 10x₁ + 14x2 x₁ + 2x₁ + 3x1 + and x₁0, x2 ≥ 0. 2x₂ ≥ 3 X₂ ≥ 4 X2 ≥ 2
Solve Example 7 by bringing x1 into the solution (instead of x2) in the initial tableau. EXAMPLE 7 Minimize x₁ + 2x2 subject to x₁ + x₂ ≥ 14 X1 X₂ ≤ 2 and x₁ ≥ 0, x2 ≥ 0.
Mark each statement True or False (T/F). Justify each answer.If row s is recessive to some other row in payoff matrix A, then row s will not be used (that is, have probability zero) in an optimal strategy for (row) player R.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.Let P be a maximizing linear programming problem and let P* be its dual. If P has an optimal solution, then the maximum value of the objective function of P in its feasible set is equal to the minimum value of the
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.If a primal linear programming problem has an optimal solution, then its dual program is bounded.
Mark each statement True or False (T/F). Justify each answer.The Minimax Theorem says that every matrix game has a solution.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.In order to begin the standard simplex method, each term in the augmented column above the horizontal line must be a nonnegative number.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.In setting up the dual linear programming problem, the matrix A in the primal problem is replaced by A–1 in the dual problem.
In Exercises 1–24, mark each statement True or False (T/F). Justify each answer.When setting up the initial simplex tableau for a canonical linear programming problem, the coefficients of the objective function go in the bottom row.
Find an example of a closed convex set S in R2 such that its profile P is nonempty but conv P ≠ S.
Mark each statement True or False (T/F). Justify each answer.Let S = {v1,...........,vk} be an affinely independent set in Rn. Then each point p in Rn has a unique representation as an affine combination of v1,........., vk.
The “B” in B-spline refers to the fact that a segment x.t / may be written in terms of a basis matrix, MS , in a form similar to a Bézier curve. That is,where G is the geometry matrix [p0 p1 p2 p3 ] and u(t) is the column vector (1, t, t2, t3). In a uniform B-spline, each segment uses the
Mark each statement True or False (T/F). Justify each answer.If {v1,..., vp} is an affinely dependent set in Rn, then the set {v1,...,vp} in R"+ of homogeneous forms may be linearly independent.
Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.
Mark each statement True or False (T/F). Justify each answer.A convex combination of points {v1; v2,.............,vk} in Rn is a linear combination of the form c1v1 + c2v2 +..........+ ck vk such that c1 + c2 + ck = 1 and ci ≥ 0 for all i .
Mark each statement True or False (T/F). Justify each answer.If v1,..., vp are in Rn and if the set of homogeneous forms {v1,...,vp} in Rn+1 is linearly independent, then {v1,..., vp} is affinely dependent.
Mark each statement True or False (T/F). Justify each answer.The cubic Bézier curve is based on four control points.
Let and and let H be the hyperplane in R4 with normal n and passing through p. Which of the points v1, v2,and v3 are on the same side of H as the origin, and which are not? P = -3 1 2 n = 5 V₁ = V2 = 1 3
Let A = {a1; a2; a3} and B = [b1; b2; b3}. Find a hyperplane H with normal n that separates A and B. Is there a hyperplane parallel to H that strictly separates A and B? 3 2 -1 a₁ = - - - - - - - [ ] - [ ] [1] = = 6, b₁ = 5
Mark each statement. Justify each answer.(T/F) The set of all affine combinations of points in a set S is called the affine hull of S.
a. Determine the number of k-faces of the 5-dimensional simplex S5 for k = 0,1, .........,4. Verify that your answer satisfies Euler’s formula.b. Make a chart of the values of fk(Sn) for n = 1,........,5 and k = 0,1,.........,4. Can you see a pattern? Guess a general formula for fk (Sn).
Mark each statement True or False (T/F). Justify each answer.If a set S is affinely independent and if p ∈ aff S, then p ∈ conv S if and only if the barycentric coordinates of p with respect to S are nonnegative.
Mark each statement True or False (T/F). Justify each answer.If v1, v2, v3, and v4 are in R3 and if the set {v2 – v1, v3 – v1, v4 – v1} is linearly independent, then {v1,..., v4} is affinely independent.
Mark each statement True or False (T/F). Justify each answer.The essential properties of Bézier curves are preserved under the action of linear transformations, but not translations.
Mark each statement. Justify each answer.(T/F) If S = {x}, then aff S is the empty set.
Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. 1 HO 2 0 1 2 -2 0 , p = 5 4 -2 2
Let x(t) and y(t) be Bézier curves from Exercise 5, and suppose the combined curve has C2 continuity (which includes C1 continuity) at p3. Set x"(1) = y"(0) and show that p5 is completely determined by p1, p2, and p3. Thus, the points p0,.......,p3 and the C2 condition determine all
Find the minimal representation of the polytope defined by the inequalities Ax ≤ b and x ≥ 0. A = 1 1 4 3 1 1 b= 18 10 28
In Exercises 7–10, let H be the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that H = [f : d]. 4 0.6. 3 2 -1 -2 [¹ 5
Mark each statement True or False (T/F). Justify each answer.Every affinely dependent set is linearly dependent.
Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. DOC −1 5 -2 2 3 3 5 0
Let x(t) and y(t) be segments of a B-spline as in Exercise 6. Show that the curve has C2 continuity (as well as C1 continuity) at x(1). That is, show that x"(1) = y"(0). This higher-order continuity is desirable in CAD applications such as automotive body design, since the curves and surfaces
Find the minimal representation of the polytope defined by the inequalities Ax ≤ b and x ≥ 0. A = 2 1 1 1 ] 1 2 b= 8 6 7
In Exercises 7–10, let H be the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that H = [f : d]. 4 HIGH -2 -2 3 7 -4 4
Mark each statement True or False (T/F). Justify each answer.Every affinely independent set is linearly independent.
Let and let H1 be the hyperplane in R4 through p1 with normal n1; and let H2 be the hyperplane through p2 with normal n2. Give an explicit description of H1∩ H2. [Find a point p in H1 ∩ H2 and two linearly independent vectors v1 and v2 that span a subspace parallel to the 2- dimensional
a. Determine the number of k-faces of the 5-dimensional hypercube C5 for k = 0,1,.........,4. Verify that your answer satisfies Euler’s formula.b. Make a chart of the values of fk(Cn) for n = 1,.........,5 and k = 0,1,........,4. Can you see a pattern? Guess a general formula for fk(Cn).
Mark each statement. Justify each answer.(T/F) If {b1,..., bk} is a linearly independent subset of Rn and if p is a linear combination of b1,...,bk, then p is an affine combination of b1,..., bk.
Mark each statement True or False (T/F). Justify each answer.Given a quadratic Bzier curve x(t) with control points p0, p1, and p2, the directed line segment p1 - P0 (from p0 to P) is the tangent vector to the curve at p0.
Mark each statement. Justify each answer.(T/F) A set is affine if and only if it contains its affine hull.
Let F1 and F2 be 4-dimensional flats in R6, and suppose that F1 ∩ F2 ≠ ∅. What are the possible dimensions of F1 ∩ F2?
Mark each statement True or False (T/F). Justify each answer.Every affine set is a convex set.
Mark each statement True or False (T/F). Justify each answer.The line segment between x and y is the set of all points of the form (1– t) x + t y, where t is in R.
Suppose v1,...........,vk are linearly independent vectors in Rn (1 ≤ k ≤ n). Then the set Xk = conv {±v1,.......,±vk} is called a k-cross polytope.a. Sketch X1 and X2.b. Determine the number of k-faces of the 3-dimensional cross polytope X3 for k = 0,1,2. What is another name for
Mark each statement True or False (T/F). Justify each answer.Given S = {b1,..., bk} in Rn, each p in aff S has a unique representation as an affine combination of b1,..., bk.
Mark each statement True or False (T/F). Justify each answer.If S = {v1,...,vp) is an affinely independent set in Rn and if p in Rn has a negative barycentric coordinate determined by S, then p is not in aff S.
Mark each statement True or False (T/F). Justify each answer.A finite set of points {v1,..., vk} is affinely dependent if there exist real numbers c1,......., ck, not all zero, such that c1+......+ ck = 1 and c1 v1 +.......+ ck vk = 0.
Mark each statement True or False (T/F). Justify each answer.When two quadratic Bézier curves with control points (p0, p1 p2} and {p2, p3, p4} are joined at p2, the combined Bézier curve will have C1 continuity at p2 if p2 is the midpoint of the line segment between p1 and p3.
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