How do we prove that a set of vectors span a plane in $\mathbb{R}^3$?
(This is not the question I am asking for help with! This is an example of the method my teacher has given us to show that a set spans something!) For instance, to prove that the set $B = \{2, x-1, x^2+1\}$ spans $\mathbb{R}_2[x]$, my teacher has had us prove the following statement
$$\exists w,y,z \in \mathbb{R}\ \mbox{such that}\ w*2+ y*(x-1) +z*(x^2+1) = ax^2 + bx + c.$$
Instead of $ax^2 + bx + c$, what would we use for a plane in $\mathbb{R}^3$?
Edit: To be more clear, the plane we have is $ax + by + cz=0$. How would I show that a set spans that?
Best Answer
The plane
$$ ax+by+cz=0$$
is the nullspace of the matrix
$$A=\left[\begin{matrix}a & b & c\end{matrix}\right]$$
Given an arbitrary set of vectors $S$, you can determine that they span the above plane by checking: