Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 0-32198-238-X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.7 Exercises - Page 62: 12


all values of $h$

Work Step by Step

The vectors are linearly dependent if and only if the matrix equation $\mathbf{A}\vec{x}=\vec{0}$ has a nontrivial solution, where $\mathbf{A}$ is the matrix whose columns are the given vectors. By row reduction, we conclude that $\begin{bmatrix}2&-6&8\\-4&7&h\\1&-3&4\end{bmatrix}\sim\begin{bmatrix}1&-3&4\\0&0&h+16\\0&0&0\end{bmatrix}$. Since the second column of the coefficient matrix contains no pivot for any value of $h$, we conclude that the solution to the homogeneous equation will always contain a free variable, i.e., that all values of $h$ leave the columns linearly dependent.
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