By monitor and I do beleive we should, I mean when someone comes to purchase a gun, he is investigated.
Do you have any idea what a full scale investigation would cost for every gun purchase? There already is a NICS (National Instant Criminal Background Check System) background check prior to a gun being sold in a gun shop. The cost of additional investigation would have to be borne by the manufacturers and sellers, if not you by your taxes. That increased cost would put firearms out of reach for the average citizen. It would also bankrupt an entire industry and put a lot of people out of work not to mention denying the average person of their Second Amendment rights.
Close the loop holes for people purchasing illegal arms.
There are no 'loopholes' for illegal firearms purchase. There are illegal sales and purchase of firearms but these are typically firearms that are stolen by a criminal and sold to a criminal. Law enforcement works hard everyday to get stolen guns off the street and out of the hands of criminals.
Stop illegal gun shops. Every gun has a #. Keep an eye on that.
The BATFE has a very strict licensing requirement and application for gun shops and monitors it very closely. I can't think of a single 'illegal' gun shop that I've seen anywhere, ever in my life. As for keeping an eye on serial numbers, that's already done. See NICS.