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Prove Theorem 6 for an affinely independent set\(S = \left\{ {{v_1},...,{v_k}} \right\}\)in\({\mathbb{R}^{\bf{n}}}\). [Hint:One method is to mimic the proof of Theorem 7 in Section 4.4.]

Short Answer

Expert verified

Theorem 6 is proved for an affinely independent set \(S = \left\{ {{v_1},...,{v_k}} \right\}\) in \({\mathbb{R}^n}\).

Step by step solution

01

Prove theorem 6

Consider the set \(S = \left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_k}} \right\}\)in\({\mathbb{R}^n}\). Let\({c_1},{c_2},...,{c_k}\), and\({d_1},{d_2},...,{d_k}\)be the scalars such that\({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_k}{{\bf{v}}_k} = {\bf{p}}\), and\({d_1}{{\bf{v}}_1} + {d_2}{{\bf{v}}_2} + ... + {d_k}{{\bf{v}}_k} = {\bf{p}}\), where\({d_1} + {d_2} + ... + {d_k} = 1\).

The systems of equations is shown below:

\({\bf{p}} = {c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_k}{{\bf{v}}_k}\)

\({\bf{p}} = {d_1}{{\bf{v}}_1} + {d_2}{{\bf{v}}_2} + ... + {d_k}{{\bf{v}}_k}\)

02

Prove Theorem 6

Subtract the equations\({\bf{p}} = {c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_k}{{\bf{v}}_k}\)and\({\bf{p}} = {d_1}{{\bf{v}}_1} + {d_2}{{\bf{v}}_2} + ... + {d_k}{{\bf{v}}_k}\)as shown below:

\(\begin{array}{l}\left( {{c_1} - {d_1}} \right){{\bf{v}}_1} + \cdots + \left( {{c_k} - {d_k}} \right){{\bf{v}}_k} = {\bf{p}} - {\bf{p}}\\\left( {{c_1} - {d_1}} \right){{\bf{v}}_1} + \cdots + \left( {{c_k} - {d_k}} \right){{\bf{v}}_k} = 0\end{array}\)

It is given that\(S = \left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_k}} \right\}\)is an independent set. Thus, all the coefficients must be 0 as shown below:

\(\begin{array}{c}{c_1} - {d_1} = 0\\{c_1} = {d_1}\end{array}\)

\(\begin{array}{c}{c_2} - {d_2} = 0\\{c_2} = {d_2}\end{array}\)

And

\(\begin{array}{c}{c_k} - {d_k} = 0\\{c_k} = {d_k}\end{array}\)

Since \({c_1} = {d_1},{c_2} = {d_2},...,{c_k} = {d_k}\), the set of scalars is unique.

Hence, Theorem 6 is proved for an affinely independent set \(S = \left\{ {{v_1},...,{v_k}} \right\}\) in \({\mathbb{R}^n}\).

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Most popular questions from this chapter

Question: 23. Let \({{\bf{v}}_1} = \left( \begin{array}{l}1\\1\end{array} \right)\), \({{\bf{v}}_2} = \left( \begin{array}{l}3\\0\end{array} \right)\), \({{\bf{v}}_3} = \left( \begin{array}{l}5\\3\end{array} \right)\) and \({\bf{p}} = \left( \begin{array}{l}4\\1\end{array} \right)\). Find a hyperplane \(f:d\) (in this case, a line) that strictly separates \({\bf{p}}\) from \({\rm{conv}}\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\{ - 6}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{7}}\\3\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{9}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{9}}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{1.4}\\{{\bf{1}}.{\bf{5}}}\\{ - {\bf{3}}.{\bf{1}}}\end{array}} \right]\), and \({\bf{x}}\left( t \right) = {\bf{a}} + t{\bf{b}}\)for \(t \ge {\bf{0}}\).Find the point where the ray\({\bf{x}}\left( t \right)\)intersects the plane that contains the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). Is this point inside the triangle?

Suppose \({\bf{y}}\) is orthogonal to \({\bf{u}}\) and \({\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to every \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\). (Hint: An arbitrary \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\) has the form \({\bf{w}} = {c_1}{\bf{u}} + {c_2}{\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to such a vector \({\bf{w}}\).)

Question 4: Repeat Exercise 2 where \(m\) is the minimum value of \(f\) on \(S\) instead of the maximum value.

In Exercises 21–24, a, b, and c are non-collinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \) and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

22. Let p be a point on the line through a and b. Show that\(det\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{p}} }\end{array}} \right] = 0\).

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