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: 40


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

Work Step by Step

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