Problem with linear transormations, their direct sums and properties

Question

Let's define linear transformation $F:V\times W\rightarrow Z$, $F_v=F(v,0)$ and $F_w(w)=F(0,w)$ where $F_v:V \rightarrow Z$ and $F_w:W \rightarrow Z.$ We have also subspace $W_0=\{0\}\times W < V \times W$. What's more $F_w$ is an isomorphism. I have two problems:

a) Show that $V \times W=W_0 \oplus \ker F$,
b) Show that transformation $h:V\rightarrow W$, $h=-F_w^{-1} \circ F_v$ fulfills condition $F(v,h(v))=0$ and show that $h$ is linear and the only one.

My several attempts:

a) To show that $V \times W=W_0 \oplus \ker F$ we ought to show that $x=a+b$ where $x\in V \times W$, $a \in \{0\}\times W$, $b\in \ker F=\{ x\in V\times W: F(x)=0\}$. Anyway, I do not know how to proceed later.

b) We want to show that $F(v,h(v))=0$.
$F(v,h(v))=F(v,-F_w^{-1}\circ F_v(v))=F(v,-F_w^{-1}(F_v(v)))=F(v,-F_w^{-1}(F(v,0)))$. I am not certain with the next step but I wonder if it is equal to $F(0,v)$ -- because $w=F_w^{-1}(F(0,w))$. Anyway it doesn't give us what we want. In addition, I don't have any idea how to show $h$ is the only one.

Answer 1

Let $G : Z \to W$ be the inverse of $F_w$.

1. Firstly, we must show that $W_0 \cap \ker(F)$ is the zero subspace. This is immediate since if we have $(x, y) \in W_0$ s.t. $F(x, y) = 0$, then we have $x = 0$ and thus $F(x, y) = F_w(y) = 0$; then $y = G(F_w(y)) = G(0) = 0$. Now, consider $(x, y) \in V \times W$. We note that $(x, y) = (x, - G(F_v(x))) + (0, y + G(F_v(x)))$ and that $(x, - G(F_v(x))) \in \ker(F)$, $(0, y + G(F_v(X))) \in W_0$. This completes the proof.
2. Suppose we have $h$ s.t. $F(v, h(v)) = 0$. Then $F_v(v) + F_w(h(v)) = 0$. Then $F_w(h(v)) = -F_v(v)$. Ten $h(v) = G(-F_v(v)) = -G(F_v(v))$. That is, $h = -G \circ F_v$. And we've already shown above that $h = -G \circ F_v$ satisfies the identity $F(v, h(v)) = 0$ for all $v$.