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: 37


W is a vector subspace of $R^3$.

Work Step by Step

Let $W=\left\{\left(x_{1}, x_{2}, 0\right) : x_{1} \text { and } x_{2} \text { are real numbers }\right\}$ and $u=(x_{1}, x_{2}, 0), v=(y_{1}, y_{2}, 0)\in W$, $c\in R$. First, one can see that $W$ contains the zero vector, and $$u+v=(x_{1}, x_{2}, 0)+(y_{1}, y_{2}, 0)=(x_{1}+y_{1}, x_{2}+y_{2}, 0)\in W,$$ $$cu=c(x_{1}, x_{2}, 0)=(cx_{1}, cx_{2}, 0)\in W.$$ Hence, W is a vector subspace of $R^3$.
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