Off-chain transactions with Coinbase
From the perspective of the Bitcoin network, Coinbase owns both the Bitcoins you transferred in, and the Bitcoins they sold to you.

This is useful from a technical perspective, because Coinbase can have as many transactions within their system as they want, and no-one outside of Coinbase needs to keep track of them.

This is problematic because Bitcoin can't give you any guarantee that Coinbase won't take your Bitcoins, and Bitcoin can't give you any guarantee that Coinbase has enough assets to pay all of their account holders. 

Coinbase could publish an audit showing how much Bitcoin all of their account holders are supposed to have, and signed statements showing control of enough Bitcoins to cover those obligations, though.