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: 20


$W$ is not a subspace of $R^2$.

Work Step by Step

Let $W$ be a subset of $V$ such that $$W=\left\{(x, y) : y=a x^{2}\right\}, \quad V=R^{2}.$$ Assume that $u=(x,ax^2), v=(y,ay^2)\in W$ and $c\in R$. Now, we have (a) $W$ contains the zero vector $(0,0)$. (b) $u+v=(x,ax^2)+(y,ay^2)=(x+y,ax^2+ay^2)=(x+y,a(x^2+y^2))$. Since $x^2+y^2\neq (x+y)^2$, then $u+v\not \in W$. Hence, $W$ is not a subspace of $R^2$.
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