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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

In Exercises 3-6, find (a) the maximum value of \(Q\left( {\rm{x}} \right)\) subject to the constraint \({{\rm{x}}^T}{\rm{x}} = 1\), (b) a unit vector \({\rm{u}}\) where this maximum is attained, and (c) the maximum of \(Q\left( {\rm{x}} \right)\) subject to the constraints \({{\rm{x}}^T}{\rm{x}} = 1{\rm{ and }}{{\rm{x}}^T}{\rm{u}} = 0\).

5. \(Q\left( x \right) = x_1^2 + x_2^2 - 10x_1^{}x_2^{}\).

Question: 4. Let A be an \(n \times n\) symmetric matrix.

a. Show that \({({\rm{Col}}A)^ \bot } = {\rm{Nul}}A\). (Hint: See Section 6.1.)

b. Show that each y in \({\mathbb{R}^n}\) can be written in the form \(y = \hat y + z\), with \(\hat y\) in \({\rm{Col}}A\) and z in \({\rm{Nul}}A\).

Question: 13. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Suppose the equation\(A{\rm{x}} = {\rm{b}}\)is consistent, and let\({{\rm{x}}^ + } = {A^ + }{\rm{b}}\). By Exercise 23 in Section 6.3, there is exactly one vector\({\rm{p}}\)in Row\(A\)such that\(A{\rm{p}} = {\rm{b}}\). The following steps prove that\({{\rm{x}}^ + } = {\rm{p}}\)and\({{\rm{x}}^ + }\)is the minimum length solution of\(A{\rm{x}} = {\rm{b}}\).

  1. Show that \({{\rm{x}}^ + }\) is in Row \(A\). (Hint: Write \({\rm{b}}\) as \(A{\rm{x}}\) for some \({\rm{x}}\), and use Exercise 12.)
  2. Show that\({{\rm{x}}^ + }\)is a solution of\(A{\rm{x}} = {\rm{b}}\).
  3. Show that if \({\rm{u}}\) is any solution of \(A{\rm{x}} = {\rm{b}}\), then \(\left\| {{{\rm{x}}^ + }} \right\| \le \left\| {\rm{u}} \right\|\), with equality only if \({\rm{u}} = {{\rm{x}}^ + }\).

Question:Find the principal components of the data for Exercise 1.

Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).

9. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A. (See the Exercises in Section 6.5.)
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