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Q8E

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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

Question: In Exercise 8, 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 \(H = \left( {f:d} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{7}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\)

  1. The normal vector is \(n = \left( {\begin{array}{*{20}{c}}4\\3\\{ - 6}\end{array}} \right)\) or a multiple
  2. The linear functional is \(f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\) , and the real number is \(d = - 8\).
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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the given data

Let \({v_1} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\), \({v_2} = \left( {\begin{array}{*{20}{c}}4\\{ - 2}\\3\end{array}} \right)\), and \({v_3} = \left( {\begin{array}{*{20}{c}}7\\{ - 4}\\4\end{array}} \right)\).

Then, the vectors are \({v_2} - {v_1} = \left( {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right),\) and \({v_3} - {v_1} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\3\end{array}} \right)\).

Step 2: Use the cross product to compute n

(a)

\(\begin{array}{c}n = \left( {{v_2} - {v_1}} \right) \times \left( {{v_3} - {v_1}} \right)\\ = \left| {\begin{array}{*{20}{c}}3&6&i\\0&{ - 2}&j\\2&3&k\end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}}0&{ - 2}\\2&3\end{array}} \right|i - \left| {\begin{array}{*{20}{c}}3&6\\2&3\end{array}} \right|j + \left| {\begin{array}{*{20}{c}}3&6\\0&{ - 2}\end{array}} \right|k\\ = 4i + 3j - 6k\end{array}\)

Thus, the normal vector is \(n = \left( {\begin{array}{*{20}{c}}4\\3\\{ - 6}\end{array}} \right)\).

Step 3: Find a linear functional f and a real number d

(b)

Using part (a), the linear functional f can be obtained as shown below:

\(\begin{array}{c}f\left( x \right) = n \cdot x\\ = \left[ {\begin{array}{*{20}{c}}4&3&{ - 6}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\end{array}\)

Note that, \({v_i}\) in \(H = \left( {f:d} \right)\) such that \(f\left( {{v_i}} \right) = d\) for \(i = 1,2,3\).

\(\begin{array}{c}d = f\left( {{v_1}} \right)\\ = f\left( {1, - 2,1} \right)\\ = 4\left( 1 \right) + 3\left( { - 2} \right) - 6\left( 1 \right)\\ = 4 - 6 - 6\\d = - 8\end{array}\)

Thus, the linear function is \(f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\) , and the real number is \(d = - 8\).

Most popular questions for Math Textbooks

A quartic Bézier curve is determined by five control points,

\({{\bf{p}}_{\bf{o}}}{\bf{,}}\,{\rm{ }}{{\bf{p}}_{\bf{1}}}\,{\bf{,}}{\rm{ }}{{\bf{p}}_{\bf{2}}}\,{\bf{,}}{\rm{ }}{{\bf{p}}_{\bf{3}}}\)and \({{\bf{p}}_4}\):

\({\bf{x}}\left( t \right) = {\left( {1 - t} \right)^4}{{\bf{p}}_0} + 4t{\left( {1 - t} \right)^3}{{\bf{p}}_1} + 6{t^2}{\left( {1 - t} \right)^2}{{\bf{p}}_2} + 4{t^3}\left( {1 - t} \right){{\bf{p}}_3} + {t^4}{{\bf{p}}_4}\) for \(0 \le t \le 1\)

Construct the quartic basis matrix \({M_B}\) for \({\bf{x}}\left( t \right)\).

Let \({v_1} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\end{array}} \right]\), \({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{5}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{3}}\end{array}} \right]\), \({p_1} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{array}} \right]\), \({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\end{array}} \right]\), \({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\end{array}} \right]\), \({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{0}}\end{array}} \right]\), \({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\), \({p_{\bf{6}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\), \({p_{\bf{7}}} = \left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{4}}\end{array}} \right]\) and let \(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of \({p_1}\), \({p_{\bf{2}}}\), and \({p_{\bf{3}}}\) with respect to S.
  3. On graph paper, sketch the triangle \(T\) with vertices \({v_1}\), \({v_{\bf{2}}}\), and \({v_{\bf{3}}}\), extend the sides as in Figure 8, and plot the points \({p_{\bf{4}}}\), \({p_{\bf{5}}}\), \({p_{\bf{6}}}\), and \({p_{\bf{7}}}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points \({p_{\bf{4}}}\), \({p_{\bf{5}}}\), \({p_{\bf{6}}}\), and \({p_{\bf{7}}}\).

Question: In Exercise 3, determine whether each set is open or closed or neither open nor closed.

3. a. \(\left\{ {\left( {x,y} \right):y > {\bf{0}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} \le y \le {\bf{3}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} < y < {\bf{3}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):xy = {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):xy \ge {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)


Prove Theorem 6 for an affinely independent set \(S = \left\{ {{v_1},...,{v_k}} \right\}\) in \({\mathbb{R}^{\bf{n}}}\). [Hint: One method is to mimic the proof of Theorem 7 in Section 4.4.]

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.]
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