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Q21E

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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 Exercises 21 and 22, mark each statement True or False. Justify each answer.

21. a. A linear transformation from \(\mathbb{R}\) to \({\mathbb{R}^n}\) is called a linear functional.

b. If \(f\) is a linear functional defined on \({\mathbb{R}^n}\) , then there exists a real number \(k\) such that \(f\left( x \right) = kx\) for all \(x\) in \({\mathbb{R}^n}\).

c. If a hyper plane strictly separates sets \(A\) and \(B\), then \(A \cap B = \emptyset \)

d. If \(A\) and \(B\) are closed convex sets and \(A \cap B = \emptyset \), then there exists a hyper plane that strictly separate \(A\) and \(B\).

  1. The given statement is false.
  2. The given statement is false.
  3. The given statement is true.
  4. The given statement is false.
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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Use the definition of linear functional

A linear functional on \({\mathbb{R}^n}\)is a linear transformation \(f\) from \({\mathbb{R}^n}\) into \(\mathbb{R}\).

So, statement (a) is false.

Step 2:  Use the concept of matrix

The system \(f\left( x \right) = Ax\) is always satisfied with a matrix \(A\) with a size \(1 \times n\) for all \(x\) in \({\mathbb{R}^n}\).

Similarly, the system \(f\left( x \right) = nx\) is always satisfied by a point \(n\) in \({\mathbb{R}^n}\), where \(x\) is also in \({\mathbb{R}^n}\).

So, statement (b) is false.

Step 3:  Use the definition of strictly separate

According to the definition of strictly separate, the common subset of the sets \(A\) and \(B\) is always null.

So, statement (c) is true.

Step 4:  Use the concept of strictly separating two sets by a hyperplane

According to the concept of strictly separating two sets by a hyperplane, if \(A\) and \(B\) are disjoint closed convex sets, but they cannot be strictly separated by a hyperplane (line in\({\mathbb{R}^2}\) ).

So, the statement in (d) is false.

Most popular questions for Math Textbooks

Question 3: Repeat Exercise 1 where \(m\) is the minimum value of f on \(S\) instead of the maximum value.

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

12.a. The essential properties of Bezier curves are preserved under the action of linear transformations, but not translations.

b. When two Bezier curves \({\mathop{\rm x}\nolimits} \left( t \right)\) and \(y\left( t \right)\) are joined at the point where \({\mathop{\rm x}\nolimits} \left( 1 \right) = y\left( 0 \right)\), the combined curve has \({G^0}\) continuity at that point.

c. The Bezier basis matrix is a matrix whose columns are the control points of the curve.

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 \({\mathbb{R}^m}\).

Question: 15. Let \(A\) be an \({\rm{m}} \times {\rm{n}}\) matrix and, given \({\rm{b}}\) in \({\mathbb{R}^m}\), show that the set \(S\) of all solutions of \(A{\rm{x}} = {\rm{b}}\) is an affine subset of \({\mathbb{R}^n}\).

Suppose that \(\left\{ {{p_1},{p_2},{p_3}} \right\}\) is an affinely independent set in \({\mathbb{R}^{\bf{n}}}\) and q is an arbitrary point in \({\mathbb{R}^{\bf{n}}}\). Show that the translated set \(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\) is also affinely independent.
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