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Q8E

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Linear Algebra and its Applications
Found in: Page 437
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

8. \(\left( {\begin{array}{{}}0\\1\\{ - 2}\\1\end{array}} \right),\left( {\begin{array}{{}}1\\1\\0\\2\end{array}} \right),\left( {\begin{array}{{}}1\\4\\{ - 6}\\5\end{array}} \right)\), \({\mathop{\rm p}\nolimits} = \left( {\begin{array}{{}}{ - 1}\\1\\{ - 4}\\0\end{array}} \right)\)

The barycentric coordinates are \(\left( {2, - 1,0} \right)\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: The barycentric coordinates

Consider the set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\)as an affinely independent. So, for every point \({\mathop{\rm p}\nolimits} \) in \({\mathop{\rm aff}\nolimits} S\), the coefficients \({c_1},...,{c_k}\) in the unique representation (7) of \({\mathop{\rm p}\nolimits} \) are referred to as barycentric coordinates (or sometimes, affine) of \({\mathop{\rm p}\nolimits} \).

Step 2: Write the augmented matrix 

Move the last row of ones to the top to simplify the arithmetic.

Write the augmented matrix as shown below:

\(\left( {\begin{array}{{}}{\widetilde {{{\bf{v}}_1}}}&{\widetilde {{{\bf{v}}_2}}}&{\widetilde {{{\bf{v}}_3}}}&{\widetilde {\bf{p}}}\end{array}} \right) \sim \left( {\begin{array}{{}}0&1&1&{ - 1}\\1&1&4&1\\{ - 2}&0&{ - 6}&{ - 4}\\1&2&5&0\\1&1&1&1\end{array}} \right)\)

Step 3: Apply row operations

Interchange row 1 and row 2.

\( \sim \left( {\begin{array}{{}}1&1&4&1\\0&1&1&{ - 1}\\{ - 2}&0&{ - 6}&{ - 4}\\1&2&5&0\\1&1&1&1\end{array}} \right)\)

At row 3, multiply row 1 by 2 and add it to row 3.

\( \sim \left( {\begin{array}{{}}1&1&4&1\\0&1&1&{ - 1}\\0&2&2&{ - 2}\\1&2&5&0\\1&1&1&1\end{array}} \right)\)

At row 4, subtract row 1 from row 4. At row 5, subtract row 1 from row 5. At row 1, subtract row 2 from row 1.

\( \sim \left( {\begin{array}{{}}1&0&3&2\\0&1&1&{ - 1}\\0&2&2&{ - 2}\\0&1&1&{ - 1}\\0&0&{ - 3}&0\end{array}} \right)\)

At row 3, multiply row 2 by 2 and subtract it from row 3.

\( \sim \left( {\begin{array}{{}}1&0&3&2\\0&1&1&{ - 1}\\0&0&0&0\\0&1&1&{ - 1}\\0&0&{ - 3}&0\end{array}} \right)\)

At row 4, subtract row 2 from row 4.

\( \sim \left( {\begin{array}{{}}1&0&3&2\\0&1&1&{ - 1}\\0&0&0&0\\0&0&0&0\\0&0&{ - 3}&0\end{array}} \right)\)

Interchange row 3 and row 5.

\( \sim \left( {\begin{array}{{}}1&0&3&2\\0&1&1&{ - 1}\\0&0&{ - 3}&0\\0&0&0&0\\0&0&0&0\end{array}} \right)\)

At row 1, multiply row 3 by 3 and subtract it from row 1.

\( \sim \left( {\begin{array}{{}}1&0&3&2\\0&1&1&{ - 1}\\0&0&1&0\\0&0&0&0\\0&0&0&0\end{array}} \right)\)

Step 4: Determine the barycentric coordinates of p

Convert the matrix into the system of equations as shown below

\(\begin{array}{}{{\mathop{\rm x}\nolimits} _1} + 3{{\mathop{\rm x}\nolimits} _3} = 2\\{{\mathop{\rm x}\nolimits} _2} + {{\mathop{\rm x}\nolimits} _3} = - 1\\{{\mathop{\rm x}\nolimits} _3} = 0\end{array}\)

Solve the system of the equation to obtain as shown below

\({{\mathop{\rm x}\nolimits} _1} = 2,{\rm{ }}{{\mathop{\rm x}\nolimits} _2} = - 1,{\rm{ }}{{\mathop{\rm x}\nolimits} _3} = 0\)

The coordinates are \(2, - 1,\,\,\,{\mathop{\rm and}\nolimits} \,\,0\), so \({\bf{p}} = 2{{\bf{v}}_1} - {{\bf{v}}_2} + 0{{\bf{v}}_3}\).

Thus, the barycentric coordinates are \(\left( {2, - 1,0} \right)\).

Most popular questions for Math Textbooks

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

10.a. If \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely dependent set in \({\mathbb{R}^n}\), then the set \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) of homogeneous forms may be linearly independent.

b. If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) and \({{\mathop{\rm v}\nolimits} _4}\) are in \({\mathbb{R}^3}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

c. Given \(S = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}} \right\}\) in \({\mathbb{R}^n}\), each \({\bf{p}}\) in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}\).

d. When color information is specified at each vertex \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) of a triangle in \({\mathbb{R}^3}\), then the color may be interpolated at a point p in \({\mathop{\rm aff}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) using the barycentric coordinates of p.

e. If T is a triangle in \({\mathbb{R}^2}\) and if a point p is on edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.

Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 25.

In Exercises 11 and 12, mark each statement True or False. Justify each answer.

12.a. The essential properties of Bezier curves are preserved under the action of linear transformations, but not translations.

b. When two Bezier curves \({\mathop{\rm x}\nolimits} \left( t \right)\) and \(y\left( t \right)\) are joined at the point where \({\mathop{\rm x}\nolimits} \left( 1 \right) = y\left( 0 \right)\), the combined curve has \({G^0}\) continuity at that point.

c. The Bezier basis matrix is a matrix whose columns are the control points of the curve.

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.]

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).
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