Let
$$A \sim B \iff \exists v \in \mathbb{R}^2 \setminus \left( \begin{array}{c} 0\\ 0\\ \end{array} \right) : \mathcal{L}_{A,v} \cap \mathcal{L}_{B,v} \neq \emptyset$$
be a relation on $\mathbb{R}^{2 \times 2}$. $\mathcal{L}_{A,b}$ is defined as $\{ x \in \mathbb{R^n}: Ax = b\}$.
Now check if $\left( \begin{array}{cc} 1 & 0\\ 0 & 0\\ \end{array} \right) $ and $\left( \begin{array}{cc} 0 & 0\\ 0 & 1\\ \end{array} \right) $ are in relation to each other.
Given all of this, my conclusion is that I need to check if $Ax = Bx$, right? Since both $x$ are the same, this means I just need to check if $A=B$?