Write a linear equation defining the subspace of R3 spanned by v1=(0,0,2) and v2=(−3,1,−1).
__________ = 0
(Write your answer in the form ax+by+cz. For example "2x+3y−4z")
I have this question to solve, but I don't have any idea how to solve this. Can anyone help me with this?
Every element of your linear subspace can be written as a linear combination of $v_1$ and $v_2$. If $(x_1,y_1,z_1)$ and $(x_2,y_2, z_3)$ 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 $$a*0 +b*0 + c*2=0$$ and $$a*(-3)+b*(1)+c*(-1)=0$$
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 $k\neq 0$