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 - Review Exercises - Page 221: 22


$W$ is not a subspace of $R^3$.

Work Step by Step

Let $W$ be a subset of $V$ such that $$W=\{(x, y, z) : x \geq 0\}, \quad V=R^{3}.$$ $W$ is not a subspace of $R^3$ since it is not closed under scalar multiplication. For example, $u=(1,2,0)\in W$ and $-2u=-2(1,2,0)=(-2,-4,0)\not \in W$.
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