Explanation Theorem 5: Given an indexed set S = { v 1 , ... , v p } in ℝ n , with p ≥ 2 , the following statements are logically equivalent. That is, either they are all true statement or they are all false. a. S is affinely dependent. B. One of the points in S is an affine combination of the other point in S . c. The set { v 2 − v 1 , ... , v P − v 1 } in ℝ n is linearly dependent. d. The set { v ˜ 1 , ... , v ˜ p } of homogeneous forms in ℝ n + 1 is linearly dependent. Theorem 8 (section 1.7) If a set contains more vectors than there are entries in each vectors, then the set is linearly dependent. That is ,any set { v 1 , ... v P } in ℝ n is linearly dependent if p > n

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

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Chapter 8.2, Problem 12E

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To show: The set of {v1,...,vp} in ℝn is affinely dependent provided p≥n+2 .

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Chapter 8.2, Problem 11E

Chapter 8.2, Problem 13E

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Chapter 8 Solutions

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

Ch. 8.1 - Plot the points v1=[10],v2=[12], v3=[31], and...Ch. 8.1 - In Exercises 14, write y as an affine combination...Ch. 8.1 - In Exercises 14, write y as an affine combination...Ch. 8.1 - In Exercises 14, write y as an affine combination...Ch. 8.1 - In Exercises 14, write y as an affine combination...Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E

Ch. 8.1 - Suppose that the solutions of an equation Ax = b...Ch. 8.1 - Prob. 11E Ch. 8.1 - a. If S = {x}, then aff S is the empty set. b. A...Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Choose a set S of three points such that aff S is...Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.2 - Describe a fast way to determine when three points...Ch. 8.2 - Prob. 2PP Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - In Exercises 9 and 10, mark each statement True or...Ch. 8.2 - a. If{v1,....,vp} is an affinely dependent set in...Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - The conditions for affine dependence are stronger...Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Let T be a tetrahedron in standard position, with...Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - In Exercises 21-24, a, b, and c are noncollinear...Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.3 - Prob. 1PP Ch. 8.3 - Let S be the set of points on the curve y = 1/x...Ch. 8.3 - Prob. 1E Ch. 8.3 - Describe the convex hull of the set S of points...Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Repeat Exercise 9 for the points q1, , q5 whose...Ch. 8.3 - Prob. 11E Ch. 8.3 - In Exercises 11 and 12, mark each statement True...Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Let v1 = [10], v2 = [12], v3 = [42], v4 = [40],...Ch. 8.3 - Prob. 16E Ch. 8.3 - In Exercises 17-20, prove the given statement...Ch. 8.3 - In Exercises 17-20, prove the given statement...Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.4 - Prob. 1PP Ch. 8.4 - Let L be the line in 2 through the points [14] and...Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - In Exercises 3 and 4, determine whether each set...Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - In Exercises 7-10, let H be the hyperplane through...Ch. 8.4 - Prob. 11E Ch. 8.4 - Let a1=[215], a2=[313], a3=[160], b1=[051],...Ch. 8.4 - Prob. 13E Ch. 8.4 - Let F1 and F2 be 4-dimensional flats in 6, and...Ch. 8.4 - In Exercises 15-20, write a formula for a linear...Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - In Exercises 15-20, write a formula for a linear...Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Let p=[41], Find a hyperplane [f : d] that...Ch. 8.4 - Let q=[23] and p=[61]. Find a hyperplane [f : d]...Ch. 8.4 - Prob. 27E Ch. 8.4 - Prob. 28E Ch. 8.4 - Prob. 29E Ch. 8.4 - Prove that the convex hull of a bounded set is...Ch. 8.5 - Find the minimal representation of the polytope P...Ch. 8.5 - Given points p1 = [10], p2 = [23], and p3 = [12]...Ch. 8.5 - Given points p1 = [01], p2 = [21], and p3 = [12]...Ch. 8.5 - Repeat Exercise 1 where m is the minimum value of...Ch. 8.5 - Repeat Exercise 2 where m is the minimum value of...Ch. 8.5 - In Exercises 5-8, find the minimal representation...Ch. 8.5 - In Exercises 5-8, find the minimal representation...Ch. 8.5 - In Exercises 5-8, find the minimal representation...Ch. 8.5 - In Exercises 5-8, find the minimal representation...Ch. 8.5 - Let S = {(x, y) : x2 + (y 1)2 1} {(3, 0)}. Is...Ch. 8.5 - Find an example of a closed convex set S in 2 such...Ch. 8.5 - Find an example of a bounded convex set S in 2...Ch. 8.5 - a. Determine the number of k-faces of the...Ch. 8.5 - a. Determine the number of k-faces of the...Ch. 8.5 - Suppose v1, , vk are linearly independent vectors...Ch. 8.5 - A k-pyramid Pk is the convex hull of a (k ...Ch. 8.5 - Prob. 16E Ch. 8.5 - In Exercises 16 and 17, mark each statement True...Ch. 8.5 - Let v be an element of the convex set S. Prove...Ch. 8.5 - If c and S is a set, define cS = {cx : x S}....Ch. 8.5 - Find an example to show that the convexity of S is...Ch. 8.5 - Prob. 21E Ch. 8.5 - Prob. 22E Ch. 8.6 - A spline usually refers to a curve that passes...Ch. 8.6 - Prob. 2PP Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Let x(t) and y(t) be Bzier curves from Exercise 5,...Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - In Exercises 11 and 12, mark each statement True...Ch. 8.6 - In Exercises 11 and 12, mark each statement True...Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - Prob. 15E Ch. 8.6 - Explain why a cubic Bzier curve is completely...Ch. 8.6 - TrueType fonts, created by Apple Computer and...Ch. 8.6 - Prob. 18E

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The set of { v 1 , ... , v p } in ℝ n is affinely dependent provided p ≥ n + 2 .

Solution Summary: The author explains that if a set contains more vectors than there are entries in each vector, then the set is linearly dependent.

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Chapter 8 Solutions