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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 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.

1.\(\left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)\)

The set of points is affinely dependent, and the relation is \(2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Condition for affinely dependent

The set is said to be affinely dependent, if the set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) in the dimension\({\mathbb{R}^n}\) exists such that \({c_1},{c_2},...,{c_p}\) not all zero, and the sum must be zero \({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

Step 2:Compute \({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}\)

Let \({{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _2} = \left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),{{\mathop{\rm v}\nolimits} _3} = \left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)\).

Compute translated points \({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}\)as shown below:

\({{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 3}\\9\end{aligned}} \right)\),

\({{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1} = \left( {\begin{aligned}{{}}{ - 1}\\3\end{aligned}} \right)\)

It is observed that two points are multiples of each other.

Step 3: Determine whether the set of points is affinely dependent

Theorem 5 states that an indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \({\mathbb{R}^n}\), with \(p \ge 2\), the following statement is equivalent. This means that either all the statements are true or all the statements are false.

  1. The set \(S\) is affinely dependent.
  2. Each of the points in \(S\) is an affine combination of the other points in \(S\).
  3. In \({\mathbb{R}^n}\), the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _1}} \right\}\)is linearly dependent.
  4. The set \(\left\{ {{{\bar v}_1},...,{{\bar v}_p}} \right\}\) of homogeneous forms in \({\mathbb{R}^{n + 1}}\) is linearly dependent.

Since two points are multiples of each other hence form a linearly dependent set.

Therefore, all statements in theorem 5 are true and thus \(S\) are affinely dependent.

\(\begin{aligned}{}{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3\left( {{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1}} \right)\\{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1} = 3{{\mathop{\rm v}\nolimits} _3} - 3{{\mathop{\rm v}\nolimits} _1}\\2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\end{aligned}\)

Thus, the set of points is affinely dependent and \(2{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2} - 3{{\mathop{\rm v}\nolimits} _3} = 0\).

Most popular questions for Math Textbooks

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 \({\rm{conv}}\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).

Question: 24. Repeat Exercise 23 for \({v_1} = \left( \begin{array}{l}1\\2\end{array} \right)\), \({v_2} = \left( \begin{array}{l}5\\1\end{array} \right)\), \({v_3} = \left( \begin{array}{l}4\\4\end{array} \right)\) and \(p = \left( \begin{array}{l}2\\3\end{array} \right)\).

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).

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 \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{0}}\end{array}} \right)\). 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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