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Chapter 7: Q23E (page 395)

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

23. \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\). (To show that \(D \cup E \subset F\), show that \(D \subset F\) and \(E \subset F\).)

Short Answer

Expert verified

It is proved that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Step by step solution

01

Set S is affine

RecallTheorem 2,whichstates that a set \(S\) is affineif and only if every affine combination of points of \(S\) lies in \(S\).

That is, \(S\) is affine if and only if \(S = {\mathop{\rm aff}\nolimits} S\).

02

Show that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\)

Since \(A \subset \left( {A \cup B} \right)\)so, it follows that \({\mathop{\rm aff}\nolimits} A \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\) according to exercise 22. Likewise, \({\mathop{\rm aff}\nolimits} B \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Therefore, \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Hence, it is proved that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

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Most popular questions from this chapter

Question: [M] A Landsat image with three spectral components was made of Homestead Air Force Base in Florida (after the base was hit by Hurricane Andrew in 1992). The covariance matrix of the data is shown below. Find the first principal component of the data, and compute the percentage of the total variance that is contained in this component.

\[S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]\]

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

15. \(\left( {\begin{aligned}{{}}{\,3}&4\\4&9\end{aligned}} \right)\)

All symmetric matrices are diagonalizable.

Compute the quadratic form \({{\bf{x}}^T}A{\bf{x}}\), when \(A = \left( {\begin{aligned}{{}}5&{\frac{1}{3}}\\{\frac{1}{3}}&1\end{aligned}} \right)\) and

a. \({\bf{x}} = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\)

b. \({\bf{x}} = \left( {\begin{aligned}{{}}6\\1\end{aligned}} \right)\)

c. \({\bf{x}} = \left( {\begin{aligned}{{}}1\\3\end{aligned}} \right)\)

In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) ,

\(\)

where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

18. Suppose \(A\) is square and invertible. Find a singular value decomposition of \({A^{ - 1}}\)
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