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Chapter 8: Q8.1-15E (page 437)

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

Short Answer

Expert verified

It is shown that the set \(S\) of all solutions of \(A{\bf{x}} = {\bf{b}}\)is an affine subset of \({\mathbb{R}^n}\).

Step by step solution

01

Describe the given statement

It is given that \(A\) it is a \(m \times n\) matrix, \({\bf{b}} \in {\mathbb{R}^m}\) and \(S\) is the set of all solutions of \(A{\bf{x}} = {\bf{b}}\).

This implies every point in \(S\) satisfies the system \(A{\bf{x}} = {\bf{b}}\), that is, \(S = \left\{ {x:A{\bf{x}} = {\bf{b}}} \right\}\).

02

 Use Theorem 3

According to theorem 3, a nonempty set \(S\) is affine if and only if it is a flat. So, we will have to show that \(S\) is a flat.

Assume that \(W\)is the set of all homogeneous solutions of the system\(A{\rm{x}} = 0\). So, \(W\)must be a subspace of \({\mathbb{R}^n}\).

03

 Use Theorem 6 of solution set of linear systems

If \(p\) be a solution of the system \(A{\bf{x}} = {\bf{b}}\), then the solution set of \(A{\bf{x}} = {\bf{b}}\) is the set of all vectors of the form \(W = p + {v_h}\), where \({v_h}\) is any solution of the homogeneous equation \(A{\bf{x}} = 0\).

Now, as \(S = W + p\), where \(p\) is a solution of the system \(A{\bf{x}} = {\bf{b}}\), so, \(S\) must be a translated set of \(W\). Thus, \(S\) is a flat.

Therefore, the set \(S\) of all solutions of \(A{\bf{x}} = {\bf{b}}\) is an affine subset of \({\mathbb{R}^n}\).

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Most popular questions from this chapter

Let \({\bf{x}}\left( t \right)\) be a B-spline in Exercise 2, with control points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\) , \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\) and determine how the derivatives \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both \({\bf{x}}'\left( 0 \right)\)and \({\bf{x}}'\left( 1 \right)\)equal 0. Justify your assertions.

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_2}\) as the origin of the coordinate system.]

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

3.\(\left( {\begin{aligned}{{}}1\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 2}\\{ - 4}\\8\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\{ - 1}\\{11}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{15}\\{ - 9}\end{aligned}} \right)\)

Question: 18. Choose a set \(S\) of four points in \({\mathbb{R}^3}\) such that aff \(S\) is the plane \(2{x_1} + {x_2} - 3{x_3} = 12\). Justify your work.

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left) {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

Question: In Exercise 3, determine whether each set is open or closed or neither open nor closed.

3. a. \(\left\{ {\left( {x,y} \right):y > {\bf{0}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} \le y \le {\bf{3}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} < y < {\bf{3}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):xy = {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):xy \ge {\bf{1}}\,\,\,and\,\,x > {\bf{0}}} \right\}\)
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