Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 4 - Vector Spaces - 4.4 Spanning Sets - Problems - Page 283: 35

Answer

See below

Work Step by Step

Given: $v_1=(1,-1,2)\\ v_2=(-2,-4,2)$ We can notice that $-2v_1=-2(1,-1,2)=(-2,2,-4)=v_2$ Obtain $av_1+(-2v_1)\\ =av_1-2bv_1\\ =(a-2b)v_1\\ \rightarrow v \in span \{v_1,v_2\}\\ \rightarrow v \in span \{v_1\}$ Hence, span $\{v_1,v_2\} \subset$ span $\{v_1\}$ Since $v=av_1 \in$ span $\{v_1\}$ $\rightarrow v \in span \{v_1,v_2\}$ $\rightarrow span\{v_1\} =\in span \{v_1,v_2\}$ Geometrically span $\{v_1,v_2\}=$ span $\{v_1\}$ is a line passing through the origin with direction $v_1$
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