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 - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 21

Answer

The set of all positive functions: $f(x)>0$ is not a vector subspace of $C(-\infty, \infty)$.

Work Step by Step

The set of all positive functions: $f(x)>0$ is not a vector subspace of $C(-\infty, \infty)$ because it is not closed under scalar multiplication. For example, $f(x)=x^2+1$ and $c=-2z\in R$. One can see that $cf(x)=-2x^2-2$ which is not a positive function.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.