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: 2


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

Work Step by Step

Let $u=(x_1,x_2,2x_1-3x_2),v=(y_1,y_2,2y_1-3y_2),\in W$ and $c\in R$ where $$W=\{(x, y, 2 x-3 y) : x \text { and } y \text { are real numbers }\}$$ Now, \begin{align*} u+v&=(x_1,x_2,2x_1-3x_2)+(y_1,y_2,2y_1-3y_2)\\ &=(x_1+y_1,x_2+y_2,2(x_1+y_1)-3(x_2+y_2)) \end{align*} which means that $u+v\in W$ and also $cu=(cx_1,cx_2,c(2x_1-3x_2))\in W$. Hence, $W$ is a vector subspace of $R^3$.
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