This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.

All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as $S=\mathbb{P}(E)$ where $E$ is a vector bundle of rank $2$ over $C$, i.e. a locally free sheaf of rank $2$ over $C$. They are always minimal, or relatively minimal, depending on your book.

An elementary transformation $S\dashrightarrow S'$ with centre $s\in S$ corresponds to blowing up $s$ and contracting the strict transform of a fibre. This gives another grs $S'$ over $C$.

That point of view which is explicit and algebraic is the one I understand and usually work with. But you can also think in the following way. Take $s\in S$ and consider the pushforward of the skyscraper sheaf $\mathbb{C}(s)$, $F=p_*\mathbb{C}(S)$. Since $s$ can be seen as $(p(s),D_s)$ where $D_s\in E_{p(s)}$ is a line at the stalk of $E$ at $p(s)$, we get a map $$u_s:E\longrightarrow F$$ which I understand it can be defined on the stalks and sends $(c,v)$ to $(c,0)$ if $c\neq p(s)$ and to $(c,v+D_s)$ otherwise. Here we identify $F_s$ with $E_s/(D_s)$ or some similar quotient. Beauville did not write $F$ as a pushforward but as the skyscraper sheaf itself. I think that is an unimportant typo. Define $E'=ker(u_s)$.

The question is:

1) $E'$ is a vector bundle of rank $2$.

2) $S'=\mathbb{P}(E')$ corresponds to the elementary transformation with centre $s$ $S\dashrightarrow S'$.

3) That transformation corresponds to the inclusion $E\rightarrow E'$.

I find hard to believe 1) is true although it must be because it is written in the book and an article by Hartshorne says it is 'easy' to see. I fail to see how the rank of $E'$ is going to be $2$ over $p(s)$. It seems to me that in the short exact sequence $$E'\longrightarrow E \longrightarrow F,$$ if you restrict to stalks you necessarily have $E'_s\cong \mathbb{C}$, but I must be being somewhat naive.

Probably the problem is that I do not understand $u_s$. I probably can do 2) and 3) once I solve 1). Any ideas?