# Is it true that $U \cap (W_1 \oplus W_2) = (U \cap W_1) \oplus (U \cap W_2)$?

Let $$W_i (i =1,2)$$ be subspaces of $$V$$. $$W_1 \cap W_2 =0$$. If $$U$$ is a subspace of $$V$$, is it true that $$U \cap (W_1 \oplus W_2) = (U \cap W_1) \oplus (U \cap W_2)$$?

If it is true, is there any way to give a formal proof? If it is not, is there a hint for constrcting a contradiction or giving a counter example?

• No, it is not true. – Charlie Frohman Jul 31 at 23:13

It is false. Take $$V=\mathbb R^2$$, $$W_1=\mathbb R\times\{0\}$$, $$W_2=\{0\}\times\mathbb R$$, and $$U=\{(x,x)\mid x\in\mathbb R\}$$. Then $$U\cap W_1=U\cap W_2=\{0\}$$, but $$U\cap(W_1\bigoplus W_2)=U\neq\{0\}$$.

• So this counterexample can be considered as a $y=x$ linear on a $x-y$ plane where $W_1$ is x-axis and $W_2$ is y-axis? – WaterBro Jul 31 at 23:16
• That's a way of looking at it, yes. – José Carlos Santos Jul 31 at 23:18