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Q7E

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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 7, 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 \(H = \left( {f:d} \right)\).

7. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{4}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{ - {\bf{2}}}\\{\bf{5}}\end{array}} \right)\)

  1. The normal vector is \(n = \left( {\begin{array}{*{20}{c}}0\\2\\3\end{array}} \right)\) or a multiple
  2. The linear function is \(f\left( x \right) = 2{x_2} + 3{x_3}\) , and the real number is \(d = 11\).
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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the given data

Let \({v_1} = \left( {\begin{array}{*{20}{c}}1\\1\\3\end{array}} \right)\), \({v_2} = \left( {\begin{array}{*{20}{c}}2\\4\\1\end{array}} \right)\), and \({v_3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 2}\\5\end{array}} \right)\).

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

Step 2: Use the cross product to compute n

(a)

\(\begin{array}{c}n = \left( {{v_2} - {v_1}} \right) \times \left( {{v_3} - {v_1}} \right)\\ = \left| {\begin{array}{*{20}{c}}1&{ - 2}&i\\3&{ - 3}&j\\{ - 2}&2&k\end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}}3&{ - 3}\\{ - 2}&2\end{array}} \right|i - \left| {\begin{array}{*{20}{c}}1&{ - 2}\\{ - 2}&2\end{array}} \right|j + \left| {\begin{array}{*{20}{c}}1&{ - 2}\\3&{ - 3}\end{array}} \right|k\\ = 0i + 2j + 3k\end{array}\)

Thus, the normal vector is \(n = \left( {\begin{array}{*{20}{c}}0\\2\\3\end{array}} \right)\).

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

(b)

Using part (a) to obtain the linear functional f as shown below:

\(\begin{array}{c}f\left( x \right) = n \cdot x\\ = \left( {\begin{array}{*{20}{c}}0&2&3\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\f\left( x \right) = 2{x_2} + 3{x_3}\end{array}\)

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

\(\begin{array}{c}d = f\left( {{v_1}} \right)\\ = f\left( {1,1,3} \right)\\ = 2\left( 1 \right) + 3\left( 3 \right)\\ = 2 + 9\\d = 11\end{array}\)

Thus, the linear function is \(f\left( x \right) = 2{x_2} + 3{x_3}\) , and the real number is \(d = 11\).

Most popular questions for Math Textbooks

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{2}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{aligned}} \right)\)

Question: 20. Let \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\) is a linear transformation, and let \(T\) be an affine subset of \({\mathbb{R}^{\bf{m}}}\), and let \(S = \left\{ {{\bf{x}} \in {\mathbb{R}^n}\,:\,f\left( {\bf{x}} \right) \in T} \right\}\). Show that \(S\) is an affine subset of \({\mathbb{R}^m}\).

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.

24. Take q on the line segment from b to c and consider the line through q and a, which may be written as \(p = \left( {1 - x} \right)q + xa\) for all real x. Show that, for each x, \(det\left[ {\begin{array}{*{20}{c}}{\widetilde p}&{\widetilde b}&{\widetilde c}\end{array}} \right] = x \cdot det\left[ {\begin{array}{*{20}{c}}{\widetilde a}&{\widetilde b}&{\widetilde c}\end{array}} \right]\). From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. (When x = 1, p = a.) When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.

In Exercises 5 and 6, let \({{\bf{b}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{ - {\bf{2}}}\end{aligned}} \right)\), and \({{\bf{b}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\\{\bf{1}}\end{aligned}} \right)\) and \(S = \left\{ {{{\bf{b}}_{\bf{1}}},\,{{\bf{b}}_{\bf{2}}},\,{{\bf{b}}_{\bf{3}}}} \right\}\). Note that S is an orthogonal basis of \({\mathbb{R}^{\bf{3}}}\). Write each of the given points as an affine combination of the points in the set S, if possible. (Hint: Use Theorem 5 in section 6.2 instead of row reduction to find the weights.)

a. \({{\bf{p}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{8}}\\{\bf{4}}\end{aligned}} \right)\)

b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{6}}\\{ - {\bf{3}}}\\{\bf{3}}\end{aligned}} \right)\)

c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{ - {\bf{1}}}\\{ - {\bf{5}}}\end{aligned}} \right)\)

Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).
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