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Q10E

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Linear Algebra and its Applications
Found in: Page 437
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

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

10.a. If \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely dependent set in \({\mathbb{R}^n}\), then the set \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) of homogeneous forms may be linearly independent.

b. If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) and \({{\mathop{\rm v}\nolimits} _4}\) are in \({\mathbb{R}^3}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

c. Given \(S = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}} \right\}\) in \({\mathbb{R}^n}\), each \({\bf{p}}\) in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}\).

d. When color information is specified at each vertex \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) of a triangle in \({\mathbb{R}^3}\), then the color may be interpolated at a point p in \({\mathop{\rm aff}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) using the barycentric coordinates of p.

e. If T is a triangle in \({\mathbb{R}^2}\) and if a point p is on edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.

  1. The given statement is False.
  2. The given statement is True.
  3. The given statement is False.
  4. The given statement is False.
  5. The given statement is True.
See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine whether the statement is True or False

a)

According to theorem 5,when the set\(S\) is affinely dependent. Then the set \(\left\{ {{{\bar v}_1},...,{{\bar v}_p}} \right\}\) of homogeneous forms in \({\mathbb{R}^{n + 1}}\) is linearly dependent.

Thus, the given statement (a) is False.

Step 2: Determine whether the statement is True or False

b)

When theset \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

Thus, the given statement (b) is True.

Step 3: Determine whether the statement is True or False

c)

Theorem 6 states that consider the set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is an affinely independent in \({\mathbb{R}^n}\). Then for everyp in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}\). Thus, it is truly only for affinely independence.

Thus, the given statement (c) is False.

Step 4: Determine whether the statement is True or False

d)

Barycentric coordinates can be used for smoothly interpolating the vertex information over the interior of a triangle. In an interpolation, the barycentric coordinates are used as weights during the linear combination of the vertex at a point.

Thus, the given statement (d) is False.

Step 5: Determine whether the statement is True or False

e)

If a point p is on the edge of the triangle, then one of the barycentric coordinates of p may be zero.

Thus, the given statement (d) is True.

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Let \({\bf{x}}\left( t \right)\) be a cubic Bézier curve determined by points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\), \({{\bf{p}}_2}\), and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\). Determine how \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have \({\bf{x}}'\left( 1 \right) = 0\)?

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_1}\) as the origin of the coordinate system.]

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.

5.\(\left( {\begin{aligned}{{}}1\\0\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\1\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 1}\\5\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\5\\{ - 3}\end{aligned}} \right)\)

Let \({\bf{x}}\left( t \right)\) and \({\bf{y}}\left( t \right)\) be cubic Bézier curves with control points \(\left\{ {{{\bf{p}}_{\bf{o}}}{\bf{,}}{{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}} \right\}\)and \(\left\{ {{{\bf{p}}_{\bf{3}}}{\bf{,}}{{\bf{p}}_{\bf{4}}}{\bf{,}}{{\bf{p}}_{\bf{5}}}{\bf{,}}{{\bf{p}}_{\bf{6}}}} \right\}\) respectively, so that \({\bf{x}}\left( t \right)\) and \({\bf{y}}\left( t \right)\) are joined at \({{\bf{p}}_3}\) . The following questions refer to the curve consisting of \({\bf{x}}\left( t \right)\) followed by \(y\left( t \right)\). For simplicity, assume that the curve is in \({\mathbb{R}^2}\).

a. What condition on the control points will guarantee that the curve has \({C^1}\) continuity at \({{\bf{p}}_3}\) ? Justify your answer.

b. What happens when \({\bf{x'}}\left( 1 \right)\) and \({\bf{y'}}\left( 1 \right)\) are both the zero vector?
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