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Chapter 8: Q2E (page 437)

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice problem 2.) If so, construct an affine dependence relation for the points.

2.\(\left( {\begin{aligned}{{}}2\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}5\\4\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 3}\\{ - 2}\end{aligned}} \right)\)

Short Answer

Expert verified

The set of points is not affinely dependent.

Step by step solution

01

Condition for the affinely dependent

The set is said to be affinely dependent if the set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) in the dimension \({\mathbb{R}^n}\) exists such that \({c_1},{c_2},...,{c_p}\) not all zero, and the sum must be zero \({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\)

02

Compute \({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}\)

Let \({{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}2\\1\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _2} = \left( {\begin{aligned}{{}}5\\4\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _3} = \left( {\begin{aligned}{{}}{ - 3}\\{ - 2}\end{aligned}} \right)\).

Compute the translated points\({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}\) as shown below

\({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}3\\3\end{aligned}} \right)\), \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 5}\\{ - 3}\end{aligned}} \right)\)

It is observed that two points are not multiples.

03

Determine whether the set of points is affinely dependent

Theorem 5states that an indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \({\mathbb{R}^n}\), with \(p \ge 2\), the following statement is equivalent. This means that either all the statements are true or all the statements are false.

  1. The set \(S\) isaffinely dependent.
  2. Each of the points in \(S\)is an affine combination of the other points in \(S\).
  3. In \({\mathbb{R}^n}\), the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _1}} \right\}\)is linearly dependent.
  4. The set \(\left\{ {{{\bar v}_1},...,{{\bar v}_p}} \right\}\) of homogeneous forms in \({\mathbb{R}^{n + 1}}\) is linearly dependent.

Since two points are not multiples, thus form a linearly independent set \(S'\).So, all statements in theorem 5 are false and \(S\) are affinely independent.

Thus, the set of points is not affinely dependent.

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Most popular questions from this chapter

In Exercises 13-15 concern the subdivision of a Bezier curve shown in Figure 7. Let \({\mathop{\rm x}\nolimits} \left( t \right)\) be the Bezier curve, with control points \({{\mathop{\rm p}\nolimits} _0},...,{{\mathop{\rm p}\nolimits} _3}\), and let \({\mathop{\rm y}\nolimits} \left( t \right)\) and \({\mathop{\rm z}\nolimits} \left( t \right)\) be the subdividing Bezier curves as in the text, with control points \({{\mathop{\rm q}\nolimits} _0},...,{{\mathop{\rm q}\nolimits} _3}\) and \({{\mathop{\rm r}\nolimits} _0},...,{{\mathop{\rm r}\nolimits} _3}\), respectively.

15. Sometimes only one-half of a Bezier curve needs further subdividing. For example, subdivision of the 鈥渓eft鈥� side is accomplished with parts (a) and (c) of Exercise 13 and equation (8). When both halves of the curve \({\mathop{\rm x}\nolimits} \left( t \right)\) are divided, it is possible to organize calculations efficiently to calculate both left and right control points concurrently, without using equation (8) directly.

a. Show that the tangent vector \(y'\left( 1 \right)\) and \(z'\left( 0 \right)\) are equal.

b. Use part (a) to show that \({{\mathop{\rm q}\nolimits} _3}\) (which equals \({{\mathop{\rm r}\nolimits} _0}\)) is the midpoint of the segment from \({{\mathop{\rm q}\nolimits} _2}\) to \({{\mathop{\rm r}\nolimits} _1}\).

c. Using part (b) and the results of Exercises 13 and 14, write an algorithm that computes the control points for both \({\mathop{\rm y}\nolimits} \left( t \right)\) and \({\mathop{\rm z}\nolimits} \left( t \right)\) in an efficient manner. The only operations needed are sums and division by 2.

Question:In Exercises 21 and 22, mark each statement True or False. Justify each answer.

21. a. A linear transformation from\(\mathbb{R}\)to\({\mathbb{R}^n}\)is called a linear functional.

b. If\(f\)is a linear functional defined on\({\mathbb{R}^n}\), then there exists a real number\(k\)such that\(f\left( x \right) = kx\)for all\(x\)in\({\mathbb{R}^n}\).

c. If a hyper plane strictly separates sets\(A\)and\(B\), then\(A \cap B = \emptyset \)

d. If\(A\)and\(B\)are closed convex sets and\(A \cap B = \emptyset \), then there exists a hyper plane that strictly separate\(A\)and\(B\).

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left) {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

a. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1} - {{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _2}} \right\}\) is linearly dependent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent. (Read this carefully.)

b. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set of homogeneous forms \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent.

c. A finite set of points \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is affinely dependent if there exist real numbers \({c_1},...,{c_k}\) , not all zero, such that \({c_1} + ... + {c_k} = 1\) and \({c_1}{{\mathop{\rm v}\nolimits} _1} + ... + {c_k}{{\mathop{\rm v}\nolimits} _k} = 0\).

d. If \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely independent set in \({\mathbb{R}^n}\) and if p in \({\mathbb{R}^n}\) has a negative barycentric coordinate determined by S, then p is not in \({\mathop{\rm aff}\nolimits} S\).

e.

If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},a,\) and \(b\) are in \({\mathbb{R}^3}\) and if ray \({\mathop{\rm a}\nolimits} + t{\mathop{\rm b}\nolimits} \) for \(t \ge 0\) intersects the triangle with vertices \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\) then the barycentric coordinates of the intersection points are all nonnegative.

Question: 2. Let Lbe the line in \({\mathbb{R}^{\bf{2}}}\) through the points \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{ - {\bf{1}}}\end{array}} \right)\). Find a linear functional f and a real number d such that \(L = \left( {f:d} \right)\).
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