Why if $U=\mathrm{span}\{0,1\}$ and $W=\mathrm{span}\{1,0\}$ then $U \cup W\neq \mathbb R^2$?

As far as I understand $\mathbb R^2=\mathrm{span}\{(1,0),(0,1)\}$ while $U \cup W=\mathrm{span}\{(1,0),(0,1)\}$. However, I saw an example that $(1,1)\notin U \cup W$.