Chapter 4 - Vector Spaces - 4.4 Spanning Sets - Problems - Page 283: 23

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Let $S=\{(x,y,z) \in R^3:x-2y-z=0\}$ We will consider vectors $v_1=(1,0,1)\\v_2=(0,1,-2)$ With any $(x,y,z) \in S \rightarrow x-2y-z=0\\ \rightarrow z=x-2y$ then $v=(x,y,x-2y)\\=(x,0,x)+(0,y,-2y)\\=x(1,0,1)+y(0,1,-2)$ Since any vector in $v\ in S $ is a linear combination of $v_1,v_2$, the set $\{v_1,v_2\}$ spans $S$.