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linear algebra and its applications

Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions

A k-pyramid Pk is the convex hull of a (k –1)-polytope Q and a point x ∉ aff Q. Find a formula for each of the following in terms of fj (Q), j = 0,.......,n – 1.a. The number of vertices of Pn: f0(Pn).b. The number of k-faces of Pn:fk (Pn), for 1 ≤ k ≤ n – 2.c. The number of (n –
Mark each statement. Justify each answer.(T/F) The affine hull of two distinct points is called a line.
Write a formula for a linear functional f and specify a number d, so that [f : d] is the hyperplane H described in the exercise.Let A be the 1 x 4 matrix [1 -3 4 –2] and let b = 5. Let H = {x in R4 : Ax = b}.
Mark each statement. Justify each answer.(T/F) A flat of dimension 1 is called a line.
Mark each statement True or False (T/F). Justify each answer.For some Rn, the dimension of a hyperplane can be the same as the dimension of a line.
Mark each statement True or False (T/F). Justify each answer.When two Bézier curves x(t) and y(t) are joined at the point where x(1) = y(0) the combined curve has G0 continuity at that point.
Mark each statement True or False (T/F). Justify each answer.A polytope is the convex hull of a finite set of points.
Solve the linear programming problem in Example 2.Data From Example 2 EXAMPLE 2 An oil refining company has two refineries that produce three grades of unleaded gasoline. Each day refinery A produces 12,000 gallons of regular, 4,000 gallons of premium, and 1,000 gallons of super, at a cost of
The Benri Company manufactures two kinds of kitchen gadgets: invertible widgets and collapsible whammies. The production process is divided into three departments: fabricating, packing, and shipping. The hours of labor required for each operation and the hours available in each department each day
Solve the linear programming problem in 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 seed, and 1 pound of bluegrass seed. Each bag of QuickGreen contains 2
In Exercises 7–10, solve the linear programming problems. Minimize subject to 5x1 + 3x₂ 2x1 + 5x2 3x1 + x₂ x₁ + 7x₂ ≥ and x₁ ≥ 0, x₂ ≥ 0. 10 6 7
In Exercises 3 – 6, find vectors b and c and matrix A so that each problem is set up as a canonical linear programming problem: Maximize cTx subject to Ax ≤ b and x ≥ 0. Do not find the solution. Maximize subject to and x₁ ≥ 0, x₂ x2 3x1 + x2 + 5x3 5x17x2 + x3 ≤ 25 2x1 + 3x2 + 4x3 =
In Exercises 3 – 6, find vectors b and c and matrix A so that each problem is set up as a canonical linear programming problem: Maximize cTx subject to Ax ≤ b and x ≥ 0. Do not find the solution. Maximize subject to 3x1 + 4x₂ - 2x3 x₁ + 2x₂ < 20 3x2 + 5x3 10 and x₁ ≥ 0, x₂ ≥ 0,
In Exercises 7–10, solve the linear programming problems. Maximize subject to 80x1 + 65x2 2x₁ + x1 + x₁ + and x₁ ≥ 0, x₂ ≥ 0. X232 x₂ 18 3x₂ ≤ 24
In Exercises 3 – 6, find vectors b and c and matrix A so that each problem is set up as a canonical linear programming problem: Maximize cTx subject to Ax ≤ b and x ≥ 0. Do not find the solution. Minimize subject to x₁ + 5x₂ - 2x3 2x₁ + x₂ + 4x3 ≤ 27 x₁6x₂ + 3x3 ≥ 40 and x₁
In Exercises 7–10, solve the linear programming problems. Maximize subject to 5x₁ + 12x₂ XI - -x1 + and x₁ ≥ 0, x₂ ≥ 0. X₂ ≤ 3 2x₂ ≤-4
In Exercises 3 – 6, find vectors b and c and matrix A so that each problem is set up as a canonical linear programming problem: Maximize cTx subject to Ax ≤ b and x ≥ 0. Do not find the solution. Minimize subject to 7x13x2 + x3 x1 - 4x2 ≥35 X2 - 2x3 = 20 and x₁ ≥ 0, x2 ≥ 0, x3 ≥ 0.
In Exercises 7–10, solve the linear programming problems. Maximize subject to 2x1 + 7x₂ -2x1 + x₂ ≤-4 x12x₂ ≤-4 and x₁ ≥ 0, x₂ ≥ 0.
In Exercises 11–14, mark each statement True or False (T/F). Justify each answer.If x̅ is an optimal solution of a canonical linear programming problem, then x is an extreme point of the feasible set.
A dog breeder decides to feed his dogs a combination of two dog foods: Pixie Power and Misty Might. He wants the dogs to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d) contained in 1 bag of each dog food are shown in the following chart, together with the
In Exercises 11–14, mark each statement True or False (T/F). Justify each answer.If a canonical linear programming problem does not have an optimal solution, then either the objective function is not bounded on the feasible set F or F is the empty set.
In Exercises 11–14, mark each statement True or False (T/F). Justify each answer.A vector x is an optimal solution of a canonical linear programming problem if f (x) is equal to the maximum value of the linear functional f on the feasible set F.
In Exercises 11–14, mark each statement True or False (T/F). Justify each answer.In a canonical linear programming problem, a nonnegative vector x is a feasible solution if it satisfies Ax ≤ b.
Betty plans to invest a total of $12,000 in mutual funds, certificates of deposit (CDs), and a high-yield savings account. Because of the risk involved in mutual funds, she wants to invest no more in mutual funds than the sum of her CDs and savings. She also wants the amount in savings to be at
Concern the subdivision of a Bézier curve shown in Figure 7. Let x(t) be the Bézier curve, with control pointsp0,........,p3, and let y(t) and z(t) be the subdividing Bézier curves as in the text, with control points q0,.................,q3 and r0,.........r3,respectively.Figure 7 From Section
Write a formula for a linear functional f and specify a number d, so that [f : d]? is the hyperplane H described in the exercise.Let H be the plane in R3 spanned by the rows of That is, H = Row B. = [ B = 3 2 54
a. Justify each equal sign:b. Show that r2 is the midpoint of the segment from p2 to p3.c. Justify each equal sign: 3.(r1 – r0) = z'(0) = .5x'(.5).d. Use part (c) to show that 8r1 = –p0 – p1 + p2 + p3 + 8r0.e. Use part (d), equation (8), and part (a) to show that r1 is the midpoint of the
Write a formula for a linear functional f and specify a number d, so that [f : d] is the hyperplane H described in the exercise.Let H be the plane in R3 spanned by the rows of B = That is, H D Row B. 1 4-5 8 -2
Write a formula for a linear functional f and specify a number d, so that [f : d] is the hyperplane H described in the exercise.Let A be the 1 x 5 matrix [2 5 -3 0 6]. Nul A is in R5. Let H = Nul A.
Mark each statement True or False (T/F). Justify each answer.When color information is specified at each vertex v1, v2, v3 of a triangle in R3, then the color may be inter- polated at a point p in aff {v1, v2, v3}using the barycentric coordinates of p.
Write a formula for a linear functional f and specify a number d, so that [f : d] is the hyperplane H described in the exercise.Let H be the column space of the matrix That is, H = Col B. B = 1 4 -7 -6 0 2
Mark each statement. Justify each answer.(T/F) A flat is a subspace.
Mark each statement True or False (T/F). Justify each answer.For some Rn, the dimension of a hyperplane can be less than the dimension of a line.
Mark each statement True or False (T/F). Justify each answer.If v1, v2, v3, a, and b are in R3 and if a ray a + tb for t ≥ 0 intersects the triangle with vertices v1, v2, and v3, then the barycentric coordinates of the intersection point are all nonnegative.
Mark each statement True or False (T/F). Justify each answer.The Bézier basis matrix is a matrix whose columns are the control points of the curve.
Mark each statement True or False (T/F). Justify each answer.A cube in R3 has exactly five facets.
Mark each statement True or False (T/F). Justify each answer.Suppose A and B are nonempty compact convex sets. Then there exists a hyperplane that strictly separates A and B if and only if A ∩ B =∅.
Mark each statement True or False (T/F). Justify each answer.Let p be an extreme point of a convex set S. If u, v ∈ S, p ∈ u̅v̅, and p ≠ u, then p = v.
Mark each statement True or False (T/F). Justify each answer.If T is a triangle in R2 and if a point p is on an edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.
Mark each statement True or False (T/F). Justify each answer.A polytope can be the convex hull of infinitely many points.
TrueType® fonts, created by Apple Computer and Adobe Systems, use quadratic Bézier curves, while PostScript® fonts, created by Microsoft, use cubic Bézier curves. The cubic curves provide more flexibility for typeface design, but it is important to Microsoft that every typeface using quadratic
Mark each statement True or False (T/F). Justify each answer.Every nonempty compact convex set S has an extreme point, and the set of all extreme points is the smallest subset of S whose convex hull is equal to S.
Mark each statement. Justify each answer.(T/F) A plane in R3 is a hyperplane.
Mark each statement. Justify each answer.(T/F) A flat of dimension 2 is called a hyperplane.
Mark each statement True or False (T/F). Justify each answer.A point p is an extreme point of a polytope P if and only if p is a vertex of P.
Mark each statement True or False (T/F). Justify each answer.If w(t) is a quadratic Bézier curve with control points p0, p1, and p2, then w(0) has the same direction as the tangent to the curve at p0 and w'(1) has the same direction as the tangent to the curve at p1.
Mark each statement True or False (T/F). Justify each answer.If S is a nonempty convex subset of Rn, then S is the convex hull of its profile.
Explain why a cubic Bézier curve is completely determined by x(0), x'(0), x(1), and x'(1).
Explain why any set of five or more points in R3 must be affinely dependent.
Show that a set {v1,..., vp} in Rn is affinely dependent when p ≥ n + 2.
Mark each statement True or False (T/F). Justify each answer.When two Bézier curves are connected with G1 geometric continuity, then the tangent vectors for the two curves at the common control point have the same direction.
Mark each statement True or False (T/F). Justify each answer.If S is a nonempty compact convex set and a linear functional attains its maximum at a point p, then p is an extreme point of S.
Mark each statement. Justify each answer.(T/F) A flat through the origin is a subspace.
Show that if {v1,v2,v3} is a basis for R3, then aff {v1,v2,v3} is the plane through v1, v2, and v3.
Suppose {v1; v2; v3} is a basis for R3. Show that Span {v2 – v1, v3 – v1} is a plane in R3. What can you say about u and v when Span {u; v} is a plane?
Use only the definition of affine dependence to show that an indexed set {v1, v2} in Rn is affinely dependent if and only if v1 = v2.
Let a. Show that the set S is affinely independent.b. c. Let T be the triangle with vertices v1, v2, and v3. When the sides of T are extended, the lines divide R2 into seven regions. See Figure 8. Signs of the barycentric coordinates of the points in each region. For example, p5 is inside the
Mark each statement True or False (T/F). Justify each answer.The 4-dimensional simplex S4 has exactly five facets, each of which is a 3-dimensional tetrahedron.
If S = {v1,.........,vk} is an affinely independent subset of Rn; prove that k ≤ n + 1.
Mark each statement. Justify each answer.(T/F) A linear transformation from R to Rn is called a linear functional.
Mark each statement. Justify each answer.(T/F) If d is a real number and f is a nonzero linear func- hyperplane in Rn. tional defined on Rn, then [f: d] is a hyperplane in Rn.
Let a. Show that the set S is affinely independent.b. Find the barycentric coordinates of p1, p2, and p3 with respect to S.c. On graph paper, sketch the triangle T with vertices v1, v2, and v3, extend the sides as in Figure 8, and plot the points p4, p5, p6, and p7. Without calculating the actual
The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed
In a, b, and c are noncollinear points in R2 and p is any other point in R2. Let Δabc denote the closed triangular region determined by a,b, and c, and let Δpbc be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that is positive, where a, b, and c
Prove Theorem 6 for an affinely independent set S = {v1.......vk} in Rn.Data From Section 8.2 Theorem 6 Let S = {V1,..., Vk} be an affinely independent set in R". Then each p in aff S has a unique representation as an affine combination of V₁,..., Vk. That is, for each p there exists a unique set
Suppose that F and G are k-dimensional flats {0 ≤ k ≤ n – 1) in Rn with F ⊆ G. Prove that F = G.
Let v be an element of the convex set S. Prove that v is an extreme point of S if and only if the set {x ∈ S : x ≠ v} is convex.
Let g(t) be defined as in Exercise 25. Its graph is called a quadratic Bézier curve, and it is used in some computer graphics designs. The points p0, p1, and p2 are called the control points for the curve. Compute a formula for g(t) that involves only p0, p1, and p2. Then show that g(t) is in conv
Prove or give a counterexample: A set S is convex if and only if for each p, q in S, the set of points of the form (1– t)p + tq, where 0 < t < 1, is contained in S.
Mark each statement True or False (T/F). Justify each answer.A 2-dimensional polytope always has the same number of vertices and edges.
Let v ∈ Rn and let k ∈ R. Prove that S = {x ∈ Rn : x · v = kg is an affine subset of Rn.
Let A be an m x n matrix and, given b in Rm, show that the set S of all solutions Ax = b is an affine subset of Rn.
If c∈ R and S is a set, define cS = {cx : x ∈ S}. Let S be a convex set and suppose c > 0 and d > 0. Prove that cS + dS = (c + d)S.
Let V be a k-dimensional subspace (0 ≤ k ≤ n – 1) of R" and let F1 = x1 + V and F = x + V for vectors x1, in R". Prove that either F1 = F2 or F1 ∩ F2 = ∅. Thus two parallel flats either coincide or are disjoint.
Choose a set S of three points such that aff S is the plane in R3 whose equation is x3 = 5. Justify your work.
Mark each statement. Justify each answer.(T/F) Given any vector n and any real number d, the set {x: n · x = d} is a hyperplane.
Let Find a hyperplane [f : d] (in this case, a line) that strictly separates p from conv {v1; v2; v3}. ¹ = ¹₂ = V₁ = -0, [H] [B] [3], and p [8] =
A polyhedron (3-polytope) is called regular if all its facets are congruent regular polygons and all the angles at the vertices are equal. Supply the details in the following proof that there are only five regular polyhedra.a. Suppose that a regular polyhedron has r facets, each of which is a
Find an example to show that the convexity of S is necessary in Exercise 25.Data From Exercise 25If c∈ R and S is a set, define cS = {cx : x ∈ S}. Let S be a convex set and suppose c > 0 and d > 0. Prove that cS + dS = .c C d/S.
Let T be a tetrahedron in “standard” position, with three edges along the three positive coordinate axes in R3, and suppose the vertices are ae1, be2, ce3, and 0, where [e1 e2 e3] = I3. Find formulas for the barycentric coordinates of an arbitrary point p in R3.
In a, b, and c are noncollinear points in R2 and p is any other point in R2. Let Δabc denote the closed triangular region determined by a,b, and c, and let Δpbc be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that is positive, where a, b, and c
Let f be a nonzero linear functional on Rn and suppose H = [f: 7]. If p = Rn, f(p) = 2, and H = H + 3p, then find d such that H1 = [f: d].
Choose a set S of four distinct points in R3 such that aff S is the plane 2x1 + x2 3x3 = 12. Justify your work.
Mark each statement. Justify each answer.(T/F) If a hyperplane strictly separates sets A and B, then A ∩ B = 0.
Mark each statement. Justify each answer.(T/F) If A and B are closed convex sets and A ∩ B = 0, then there exists a hyperplane that strictly separates A and B.
Let Find a hyperplane [f : d] that strictly separates A and B. - [i]· 4 = [3] · 4 P = A = {x: ||xp|| ≤ 1},
If A and B are convex sets, prove that A + B is convex.
Let {p1, p2; p3} be an affinely dependent set of points in Rn and let f : Rn → Rm be a linear transformation. Show that {f(p1), f(p2), f(p3)} is affinely dependent in Rm.
Letand let Find a hyperplane [f : d] that strictly separates A and B. P= [1] let A = {x ||x|| ≤ 3},
Let S be an affine subset of Rn, suppose f: Rn → Rm is a linear transformation, and let f(S) denote the set of images {f(x) : x ∈ S}. Prove that f(S) is an affine subset of Rm.
Let V be an (n – 1)-dimensional subspace of R" and suppose p ∈ Rn but p ∉ V. Prove that each vector x in Rn has a unique representation as x = v + cp, where v ∈ V and c ∈ R.
If m is the maximum value of the linear functional f on the convex set S, and p, q are points in S such that f(p) = f(q) = m, show that f(x) = m for all x in p̅q̅.
Mark each statement. Justify each answer.(T/F) If there exists a hyperplane H such that H does not strictly separate two sets A and B, then (conv A) ∩ (conv B) ≠ 0.
Suppose that {p1,p2, p3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {p1 + q, p2 + q, p3 + q} is also affinely independent.
If B(p. δ) is the open ball with center p and radius 8 in R", prove that λB(p, δ) = B(λp, λδ), where δ > 0 and λ > 0. This means that dilations and nonzero contractions map circles in R2 onto circles and balls in Rn onto balls.
Let f : Rn → Rm be a linear transformation, let T be an affine subset of Rm, and let S = {x ∈ Rn : f (x) ∈ T}. Show that S is an affine subset of Rn.
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text). If A ⊆ B and B is affine, then aff A ⊆ B.
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).Find an example in R2 to show that equality need not hold in the statement of Exercise
Deal with the following concepts: A point p is called a positive combination of the points v1,........., vk if p = c1v1 +..........+ckvk, with all ci ≥ 0. The set of all positive combinations of points of a set S is called the positive hull of S and is denoted by pos S.Let S be a
Prove the given statement about subsets A and B of Rn, or provide the required example in R2. A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).[(aff A) ∪ (aff B)] ⊆ aff (A ∪ B). To show that D ∪ E ⊆ F, show that D ⊆ F and

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