Let put it this way : when you enter a dojo, an aikido dojo for exemple, why do you bow ? From what I have gathered, you bow is an aknowledgement that a dojo is a space where usual rules do not apply.
First, il Japanese culutre, a dojo is something of an hallowed ground (please remember that Japansese notion of holy is quite far off the monotheist one). So you bow necouse you enter a sacred place.
Second, usual social distinctions do not apply in a dojo. The distinctions are based on rank an time of practice, and not on wealth or political power. Basically, it means a 25 years-old student can command respect from a 45-years old CEO.
Third, the usual rules of behavior no not apply as well. On a mat, you are not personnally offended by the fact that someone is trying to hit you (it is neseccary for the practice), and you will allow people to have contact much more intimate that what you would tolerate elsewere.
Hence, bowing is an aknowlegdement of those three points. Hence, I think it is right to bow to any mat you will be practicing (if you don't go aigainst local customs bu bowing), or even to the patch of grass upon which you will try you ukemis.
For those reading French, I wrote an article about the significance of the Dojo as a special place. I am no expert, so errors are probably numerous, but I tried to summarize what I learned from various sources.