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Chapter 8: Q8.1-12E (page 437)

Question: 12. In Exercises 11 and 12, mark each statement True or False. Justify each answer.

a. If \(S = \left\{ {\bf{x}} \right\}\), then \({\rm{aff}}\,S\) is the empty set.

b. A set is affine if and only if it contains its affine hull.

c. A flat of dimension 1 is called a line.

d. A flat of dimension 2 is called a hyper plane.

e. A flat through the origin is a subspace.

Short Answer

Expert verified
  1. The given statement is False.
  2. The given statement is True.
  3. The given statement is True.
  4. The given statement is False.
  5. The given statement is True.

Step by step solution

01

Use the definition of affine combination

According to the definition of affine combination, the set of all affine combinations of points in a set S is called the affine hull (or affine span) of \(S\), denoted by \({\rm{aff }}S\).

So, the statement (a) is False.

02

Use theorem 2

According to theorem 2,a set \(S\) is affine if and only if every affine combination of points of \(S\) lies in \(S\).

That is, \(S\) is affine if and only if \(S = {\rm{aff}}S\).

So, statement (b) is True.

03

Use the concept of proper subspaces in \({\mathbb{R}^3}\)

The proper flats in \({\mathbb{R}^3}\) are points (zero-dimensional), lines (one-dimensional), and planes, but not hyperplanes (two-dimensional), which may or may not pass through the origin.

So, statement in (c) is True, whereas (d) is False.

04

Use the concept of flat

A flat through the origin is a subspace only, which is translated by the 0 vectors.

So, statement in (e) is True.

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Most popular questions from this chapter

Explain why any set of five or more points in \({\mathbb{R}^3}\) must be affinely dependent.

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 \(L = \left( {f:d} \right)\).

Question: 23. Let \({{\bf{v}}_1} = \left( \begin{array}{l}1\\1\end{array} \right)\), \({{\bf{v}}_2} = \left( \begin{array}{l}3\\0\end{array} \right)\), \({{\bf{v}}_3} = \left( \begin{array}{l}5\\3\end{array} \right)\) and \({\bf{p}} = \left( \begin{array}{l}4\\1\end{array} \right)\). Find a hyperplane \(f:d\) (in this case, a line) that strictly separates \({\bf{p}}\) from \({\rm{conv}}\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).

TrueType fonts, created by Apple Computer and Adobe Systems, use quadratic Bezier curves, while PostScript fonts, created by Microsoft, use cubic Bezier curves. The cubic curves provide more flexibility for typeface design, but it is important to Microsoft that every typeface using quadratic curves can be transformed into one that used cubic curves. Suppose that \({\mathop{\rm w}\nolimits} \left( t \right)\) is a quadratic curve, with control points \({{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},\) and \({{\mathop{\rm p}\nolimits} _2}\).

  1. Find control points \({{\mathop{\rm r}\nolimits} _0},{{\mathop{\rm r}\nolimits} _1},{{\mathop{\rm r}\nolimits} _2},\), and \({{\mathop{\rm r}\nolimits} _3}\) such that the cubic Bezier curve \({\mathop{\rm x}\nolimits} \left( t \right)\) with these control points has the property that \({\mathop{\rm x}\nolimits} \left( t \right)\) and \({\mathop{\rm w}\nolimits} \left( t \right)\) have the same initial and terminal points and the same tangent vectors at \(t = 0\)and\(t = 1\). (See Exercise 16.)
  1. Show that if \({\mathop{\rm x}\nolimits} \left( t \right)\) is constructed as in part (a), then \({\mathop{\rm x}\nolimits} \left( t \right) = {\mathop{\rm w}\nolimits} \left( t \right)\) for \(0 \le t \le 1\).

Question 1: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\3\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right)\) in \({\mathbb{R}^2}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a.\(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 3{x_1} + {x_2}\)
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