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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

The “B” in B-spline refers to the fact that a segment \({\bf{x}}\left( t \right)\)may be written in terms of a basis matrix, \(\,{M_S}\) , in a form similar to a Bézier curve. That is,

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TrueType fonts, created by Apple Computer and Adobe Systems, use quadratic Bezier curves, while PostScript fonts, created by Microsoft, use cubic Bezier curves. The cubic curves provide more flexibility for typeface design, but it is important to Microsoft that every typeface using quadratic curves can be transformed into one that used cubic curves. Suppose that \({\mathop{\rm w}\nolimits} \left( t \right)\) is a quadratic curve, with control points \({{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},\) and \({{\mathop{\rm p}\nolimits} _2}\).

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