Show that $\mathbb R^3$ is the direct sum of two of its subspaces

Question

Let $U$ and $V$ be finite dimensional subspaces of $\mathbb R^3$ such that $\dim (U)=1$ and $\dim (V)=2$. Want to show that $\mathbb R^3$ is the direct sum of $U$ and $V$. How can we show that $\dim(U \cap V)=0$?

Answer 1

You can't because it doesn't have to be true. Suppose that $$U=\{(x,0,0)\,|\,x\in\mathbb{R}\}$$and that$$W=\{(x,y,0)\,|\,x,y\in\mathbb{R}\}.$$