Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 13E from Chapter 1.7 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 13E
Chapter:
Problem:
In Exercises 11–14, find the value(s) of h for which the vectors are linearly dependent. Justify each answer.
Step-by-Step Solution
Given Information
We are given with three vectors \[{{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\5\\{ - 3}\end{array}} \right],{{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 9}\\6\end{array}} \right],{\rm{ }}\,\,{\rm{and}}\,\,{\rm{ }}{{\bf{v}}_3} = \left[ {\begin{array}{*{20}{c}}3\\h\\{ - 9}\end{array}} \right]\]We have to find the value of $h$ fro which the three vectors are linearly dependent
Step-1: The Augmented Matrix
For linear dependencies \[{x_1}{{\bf{v}}_1} + {x_2}{{\bf{v}}_2} + {x_3}{{\bf{v}}_3} = 0\]The Augmented Matrix\[\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\5&{ - 9}&h&0\\{ - 3}&6&{ - 9}&0\end{array}} \right]\]
Step-2: Row-Reduced Augmented matrix
\[\begin{array}{l}\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\5&{ - 9}&h&0\\{ - 3}&6&{ - 9}&0\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\0&1&{h - 15}&0\\0&0&0&0\end{array}} \right]::\,\,\left\{ \begin{array}{l}{R_2} = {R_2} - 5{R_1}\\{R_3} = {R_3} + 3{R_1}\end{array} \right\}\end{array}\]Since the last row has all elements 0, for all values of h, the vectors are linearly dependent.
For All Values of $h$, the vectors are linearly dependent
We are given with three vectors \[{{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\5\\{ - 3}\end{array}} \right],{{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 9}\\6\end{array}} \right],{\rm{ }}\,\,{\rm{and}}\,\,{\rm{ }}{{\bf{v}}_3} = \left[ {\begin{array}{*{20}{c}}3\\h\\{ - 9}\end{array}} \right]\]We have to find the value of $h$ fro which the three vectors are linearly dependent
Step-1: The Augmented Matrix
For linear dependencies \[{x_1}{{\bf{v}}_1} + {x_2}{{\bf{v}}_2} + {x_3}{{\bf{v}}_3} = 0\]The Augmented Matrix\[\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\5&{ - 9}&h&0\\{ - 3}&6&{ - 9}&0\end{array}} \right]\]
Step-2: Row-Reduced Augmented matrix
\[\begin{array}{l}\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\5&{ - 9}&h&0\\{ - 3}&6&{ - 9}&0\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&3&0\\0&1&{h - 15}&0\\0&0&0&0\end{array}} \right]::\,\,\left\{ \begin{array}{l}{R_2} = {R_2} - 5{R_1}\\{R_3} = {R_3} + 3{R_1}\end{array} \right\}\end{array}\]Since the last row has all elements 0, for all values of h, the vectors are linearly dependent.
For All Values of $h$, the vectors are linearly dependent