Prove or disprove

$\displaystyle W, U_1, U_2$ are subspaces of $\displaystyle V$.

If $\displaystyle V=U_1 \oplus U_2$

then $\displaystyle W = (U_1 \cap W) \oplus (U_2 \cap W)$

Attempt:

False

let's say $\displaystyle dimV = 8, dimU_1 = dimU_2 = 4 (dimV = dimU_1 + dimU_2)$

and $\displaystyle dimW = 5$

then $\displaystyle dim(U_1 \cap W)$ may equal 4

and $\displaystyle dim(U_2 \cap W)$ may equal 4

and then we get $\displaystyle 5 \neq 4+4$