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: 18


$W$ is not a vector subspace of $M_{n,n}$.

Work Step by Step

Let $A, B$ be two $n\times n$ matrices in $W$. Then, $A^2=A$ and $B^2=B$. Now, $$(A+B)^2=A^2+B^2+AB+BA\\ \hspace{1.4cm}=A+B+AB+BA$$ which means that $A+B$ does not belong to $W$. Hence, $W$ is not a vector subspace of $M_{n,n}$.
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.