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: 11



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 $\begin{bmatrix}1&3&-1\\-1&-5&5\\4&7&h\end{bmatrix}\sim\begin{bmatrix}1&3&-1\\0&1&-2\\0&0&h-6\end{bmatrix}$. If there is a pivot in every column of the coefficient matrix, then the homogeneous system has only the trivial solution. To avoid this, we can set $h=6$ so that the final row is all zeros.
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