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: 14



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 $\begin{bmatrix}1&-5&1\\-1&7&1\\3&8&h\end{bmatrix}\sim\begin{bmatrix}1&-5&1\\0&1&1\\0&0&h-26\end{bmatrix}$. Since the homogeneous matrix equation has a nontrivial solution if and only if at least one column of the coefficient matrix lacks a pivot, we set $h=26$, making the final row all zeros and providing a free variable in the third column.
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