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Chapter 5: Q12E (page 267)

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

a. A set is convex if \({\bf{x}},{\bf{y}} \in S\) implies that the line segment between x and y is contained is S.

b. If S and T are convex sets, then \(S \cap T\) is also convex.

c. If S is a nonempty subset of \({\mathbb{R}^{\bf{5}}}\) and \({\bf{y}} \in {\bf{conv}}\,\,S\), then there exists distinct points \({{\bf{v}}_{\bf{1}}}\),鈥�.\({{\bf{v}}_{\bf{6}}}\).

Short Answer

Expert verified

a. The given statement is True.

b. The given statement is True

c. The given statement is True.

Step by step solution

01

Check for statement (a)

Using the definition of convex, a set S is convex if for each x, yin S, the line \(\overline {xy} \) is contained in S.

So, the given statement is True.

02

Check for statement (b)

From theorem 8, if S and Trepresent a convex set, their intersection is also a convex set.

So, the given statement is True.

03

Check for statement (c)

According to theorem 10, every point in the set in \({\mathbb{R}^n}\), then every point in conv S can be represented as a convex combination. So, y can be expressed as,

\(y = {c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ..... + {c_6}{{\bf{v}}_6}\)y=

It is a convex combination of \({{\bf{v}}_1}\), \({{\bf{v}}_2}\),鈥�.,\({{\bf{v}}_6}\).

So, the statement is True.

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Most popular questions from this chapter

Question 19: Let \(A\) be an \(n \times n\) matrix, and suppose A has \(n\) real eigenvalues, \({\lambda _1},...,{\lambda _n}\), repeated according to multiplicities, so that \(\det \left( {A - \lambda I} \right) = \left( {{\lambda _1} - \lambda } \right)\left( {{\lambda _2} - \lambda } \right) \ldots \left( {{\lambda _n} - \lambda } \right)\) . Explain why \(\det A\) is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.)

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. The determinant of \(A\) is the product of the diagonal entries in \(A\).
  2. An elementary row operation on \(A\) does not change the determinant.
  3. \(\left( {\det A} \right)\left( {\det B} \right) = \det AB\)
  4. If \(\lambda + 5\) is a factor of the characteristic polynomial of \(A\), then 5 is an eigenvalue of \(A\).

Exercises 19鈥�23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

20. Let \(p\left( t \right){\bf{ = }}\left( {t{\bf{ - 2}}} \right)\left( {t{\bf{ - 3}}} \right)\left( {t{\bf{ - 4}}} \right){\bf{ = - 24 + 26}}t{\bf{ - 9}}{t^{\bf{2}}}{\bf{ + }}{t^{\bf{3}}}\). Write the companion matrix for \(p\left( t \right)\), and use techniques from chapter \({\bf{3}}\) to find the characteristic polynomial.

(M)The MATLAB command roots\(\left( p \right)\) computes the roots of the polynomial equation \(p\left( t \right) = {\bf{0}}\). Read a MATLAB manual, and then describe the basic idea behind the algorithm for the roots command.

Question: In Exercises \({\bf{5}}\) and \({\bf{6}}\), the matrix \(A\) is factored in the form \(PD{P^{ - {\bf{1}}}}\). Use the Diagonalization Theorem to find the eigenvalues of \(A\) and a basis for each eigenspace.

6. \(\left( {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{{\bf{ - 2}}}\\{\bf{2}}&{\bf{5}}&{\bf{4}}\\{\bf{0}}&{\bf{0}}&{\bf{5}}\end{array}} \right){\bf{ = }}\left( {\begin{array}{*{20}{c}}{{\bf{ - 2}}}&{\bf{0}}&{{\bf{ - 1}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}\\{\bf{1}}&{\bf{0}}&{\bf{0}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{5}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&4\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{0}}&{\bf{0}}&{\bf{1}}\\{\bf{2}}&{\bf{1}}&{\bf{4}}\\{{\bf{ - 1}}}&{\bf{0}}&{{\bf{ - 2}}}\end{array}} \right)\)
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