Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.4 Spanning Sets and Linear Independence - 4.4 Exercises - Page 178: 38


$S$ is not linearly independent.

Work Step by Step

It is eas to note that $$(1,5,-3)=(1,0,0)+\frac{5}{4}(0,4,0)+\frac{1}{2}(0,0,-6).$$ Then we have the nontrivial combination $$(1,5,-3)-(1,0,0)-\frac{5}{4}(0,4,0)-\frac{1}{2}(0,0,-6)=(0,0,0,0).$$ Which means that $S$ is not linearly independent.
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