# Write a linear equation defining the subspace of R_3 spanned

Write a linear equation defining the subspace of ${R}_{3}$ spanned by ${v}_{1}=\left(0,0,2\right)$ and ${v}_{2}=\left(-3,1,-1\right)$.
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lilwayne10j6o
Every element of your linear subspace can be written as a linear combination of ${v}_{1}$ and ${v}_{2}$. If $\left({x}_{1},{y}_{1},{z}_{1}\right)$ and $\left({x}_{2},{y}_{2},{z}_{3}\right)$ both satisfy
$ax+by+cz=0$
then so does any linear combination of the two. Since ${v}_{1},{v}_{2}$ span a 2 dimensional subspace of a 3 dimensional space, we only need one equation to describe it, and both ${v}_{1}$ and ${v}_{2}$ must solve it. So we need
$a\star 0+b\star 0+c\star 2=0$
and
$a\star \left(-3\right)+b\star \left(1\right)+c\star \left(-1\right)=0$
Solving these equations gives (a,b,c) up to some factor that we can choose, since
$ax+by+cz=0$
and
$kax+kby+kcz=0$
have the same solution set if $k\ne 0$