Give an example of $W \neq(U_1 \cap W) \oplus (U_2 \cap W)$ if $V=U_1 \oplus U_2$ and $U_1,U_2,W$ are the subspaces of $V$

Question

Give an example of $W \neq (U_1 \cap W) \oplus (U_2 \cap W)$ if $V=U_1 \oplus U_2$ and $U_1,U_2,W$ are the subspaces of $V$

Let $U_1=\{(x,0)\}$ be a subspace of $\mathbb{R^2}$ over $\mathbb R$.

Let $U_2=\{(0,y)\}$ be a subspace of $\mathbb{R^2}$ over $\mathbb R$.

Let $W=\{(x,y)\}$ be a subspace of $\mathbb{R^2}$ over $\mathbb R$.

Then: $$ (U_1 \cap W) \oplus (U_2 \cap W)=((x,0) \cap(x,y))\oplus((0,y)\cap(x,y))=\\ (\emptyset)\oplus(\emptyset)=\emptyset \neq W $$

This seems a too easy example and I feel like I'm missing something. Am I in the correct direction?

Answer 1

The classic example of this is having $U_1,U_2,W$ be three distinct lines through the origin in a space of dimension$~2$ (both intersections are reduced to the zero vector, but $W$ is not).