Resolving indeterminacy of map on projective spaces induced by a linear map Let $V$ and $W$ be two $k$-vector spaces, where $k$ can be assumed as $\mathbb{C}$, and 
$f:V\longrightarrow W$ a non null $k$-linear map with kernel $K$. The map $f$ naturally induces a rational morphism $F:\mathbb{P}(V)\dashrightarrow\mathbb{P}(W)$, where $F$ is defined in the non-empty open set $\mathbb{P}(V)\backslash\mathbb{P}(K)$. Then via blowing up we resolve the map 
$F$, that is. Let $\pi: Bl_{\mathbb{P}(K)}(\mathbb{P}(V))\longrightarrow\mathbb{P}(V)$ be the blow up of $\mathbb{P}(V)$ along $\mathbb{P}(K)$, and $F':Bl_{\mathbb{P}(K)}(\mathbb{P}(V))\longrightarrow\mathbb{P}(W)$ be the map which resolves $F$ or more precisely $F = F'\circ\pi$.
I would like to know two things and any help by indicating books, solution any hint would let me happy.
1 - Is it simple to know what is the blow up in this specific case?
2- What is the pull back $(F')^*\mathcal{O}_{\mathbb{P}(W)}(1)$?
Thank you so much for any help!! 

Let me assume $f$ is surjective (otherwise, replace $W$ by $Im(f)$). Then
$$
Bl_{\mathbb{P}(K)}(\mathbb{P}(V)) = 
\mathbb{P}_{\mathbb{P}(W)}(K \otimes \mathcal{O} \oplus \mathcal{O}(-1)).
$$
Next, if $H_V$ and $H_W$ are the pullbacks of the hyperplane classes of $\mathbb{P}(V)$ and $\mathbb{P}(W)$, respectively, and $E$ is the exceptional divisor of the blowup, then
$$
H_W = H_V - E.
$$
