Is the space of matrices with rank less than 2 a subspace?

Question

Let $W = \{A ∈ M_{2\times2}(\mathbb{R}) : \mathrm{rank}(A) < 2\}$.
Is $W$ a subspace of $M_{2\times 2}(\mathbb{R})$?

Answer 1

No. Consider $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}0&0\\0&1\end{pmatrix}$.

Then $A,B\in W$ but $A+B=I\not\in W$ since $\mathrm{rank}(I)=2$.