Prove that $(V \oplus W) / (X \oplus Y)$ is isomorphic to $(V / X) \oplus (W / Y)$

Question

Let V and W be two vector spaces over the field F. Let X ⊆ V and Y ⊆ W be subspaces. I don't understand how isomorphism figures here. Any help would be appreciated.

Answer 1

Every element of $(V/X)\oplus (W/Y)$ has a form $(v+X,w+Y)$. Consider a map $$(V/X\oplus W/Y)\to (V\oplus W)/(X\oplus Y): (v+X,w+Y)\to (v,w)+X\oplus Y$$ and prove its correctness and bijectivity.

Answer 2

Let $x$ be given in $V\oplus W$ and write $x=(v,w)$ for $v\in V,w\in W$ and further write $v$ and $w$ as $v=p_X(v)+v'$ and $w=p_Y(w)+w'$ where $p_X$ is the projection from $V$ onto $X$ and $p_Y$ is the projection of $W$ onto $Y$ (this is a unique representation).

Let $f\colon V\oplus W\rightarrow (V/X)\oplus (W/Y)$ be given by $$f(x)=(v'+X,w'+Y).$$ Show that this map is linear. Further, show that the kernel of this map is $X\oplus Y$, and $f$ is surjective. Hence, using the first isomorphism theorem, show that $$(V\oplus W)/(X\oplus Y)=(V\oplus W)/\ker f\cong\mbox{Im }f= (V/X)\oplus (W/Y).$$