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Q11E

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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

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

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

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: State the condition for affinely dependence

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 for non-zero scalars\({c_1},{c_2},...,{c_p}\), the sum of scalars is zero i.e.\({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

Step 2: Show affinely dependence

Consider the set of five points \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4},{{\bf{v}}_5}} \right\}\).

Let \(\left\{ {{{\bf{v}}_2} - {{\bf{v}}_1},{{\bf{v}}_3} - {{\bf{v}}_1},{{\bf{v}}_4} - {{\bf{v}}_1},{{\bf{v}}_5} - {{\bf{v}}_1}} \right\}\) be the set of vectors in \({\mathbb{R}^3}\).This new set of four points must be linearly dependent when a set of vectors is translated by eliminating the first point or any other point.

From \({\mathbb{R}^3}\), the number of entries is 3. From the above-considered set of points, the number of vectors is 4. That is, \(n = 3\), and \(p = 4\).

Here, the set contains more vectors or points than the number of entries. So,\(p > n\)and the original set of five points \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4},{{\bf{v}}_5}} \right\}\) is affinely dependent.

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

Most popular questions for Math Textbooks

In Exercises 21–24, a, b, and c are noncollinear points in \({\mathbb{R}^{\bf{2}}}\) and p is any other point in \({\mathbb{R}^{\bf{2}}}\). Let \(\Delta {\bf{abc}}\) denote the closed triangular region determined by a, b, and c, and let \(\Delta {\bf{pbc}}\) be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that \(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\) is positive, where \(\overrightarrow {\bf{a}} \), \(\overrightarrow {\bf{b}} \) and \(\overrightarrow {\bf{c}} \) are the standard homogeneous forms for the points.

23. Let p be any point in the interior of \(\Delta {\bf{abc}}\), with barycentric coordinates \(\left( {r,s,t} \right)\), so that

\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\left[ {\begin{array}{*{20}{c}}r\\s\\t\end{array}} \right] = \widetilde {\bf{p}}\)

Use Exercise 21 and a fact about determinants (Chapter 3) to show that

\(r = \left( {area of \Delta pbc} \right)/\left( {area of \Delta abc} \right)\)

\(s = \left( {area of \Delta apc} \right)/\left( {area of \Delta abc} \right)\)

\(t = \left( {area of \Delta abp} \right)/\left( {area of \Delta abc} \right)\)

Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 25.

Question: Let \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{0}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and let H be the hyperplane in\({\mathbb{R}^{\bf{4}}}\) with normal n and passing through p. Which of the points \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\) are on the same side of H as the origin, and which are not?

Question: In Exercise 9, let H be the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

9. \(\left( {\begin{array}{*{20}{c}}1\\0\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}2\\3\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\2\\2\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\1\\1\\1\end{array}} \right)\)

Question: 2. Let L be 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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