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: 39


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

Work Step by Step

Let $W=\{(a, b, a+2 b) \cdot a \text { and } b \text { are real pumbers }\}$, $u=(x_1, x_2, x_1+2 x_2),v=(y_1, y_2, y_1+2 y_2)\in W$, $c\in R$. Now, $W$ contains the zero vector and \begin{align*} u+v&=(x_1, x_2, x_1+2 x_2)+(y_1, y_2, y_1+2 y_2)\\ &= (x_1+y_1,x_2+ y_2, x_1+y_1+2( x_2+y_2))\in W.\\ \end{align*} Also, \begin{align*} cu&=c(x_1, x_2, x_1+2 x_2)\\ &=(cx_1, cx_2, c(x_1+2 x_2))\in W.\\ \end{align*} Hence, $W$ is a vector subspace of $R^3$.
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