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: 29


$S$ is linearly independent set.

Work Step by Step

Assume the combination $$a(-2,2)+b(3,5)=(0,0).$$ We have the system \begin{align*} -2a+3b&=0\\ 2a+5b&=0. \end{align*} Since the determinant of the coefficient matrix is given by $$\left| \begin{array} {cc} -2&3\\2&5 \end{array} \right|=-16\neq 0$$ then the system has unique solution, that is, $$a=0, \quad b=0.$$ Consequently, $S$ is linearly independent set.
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