# Find :$W_1 \cap (W_2 +W_3)$

let $$W_1, W_2$$ and $$W_3$$ be subspaces of $$V$$

Let $$V=\mathbb R^2$$ and $$W_1=\{(x,y)\in V:x=y\}$$ $$W_2=\{(x,y)\in V:x=0\}$$ $$W_3=\{(x,y)\in V: y=0\}$$

Then $$W_1 \cap (W_2 +W_3)$$ is equal ______?

My attempt:
$$W_1 \cap (W_2 +W_3)= (x,x) \cap \{(x,0) +(0,y)\}=(x,x) \cap(x,y) =(x,x)$$

Is it correct?

Any hints/solution?

Thank you!

You only have to prove that $$W_2+W_3=\Bbb R^2$$, then $$W_1\cap(W_2+W_3)=W_1$$ immediately follows.