I wrote a little text that outlines why <a href="/wp-content/uploads/2013/04/vector-bundles.pdf">vector bundles and locally free sheaves</a> are the same thing. This approach is very messy with a lot of gluing, mostly because I did not look at Exercise II.5.18 in Hartshorne right away. The construction given there is much more canonical and preferable over mine. However, I decided to put this online simply <i>because</i> it is different and personally, it gave me a better <i>feeling</i> for why the two notions coincide.