Press, and hold you're PS3 on button. Until it beeps twice *I think* That is pretty much resetting setting to factory. Plug the hdmi into TV/Monitor select what you like.
Put the PS3 off into Standby mode (red light). Press the power button to turn it on but don't let go until it beeps a second time. It will then cycle through the highest resolution settings for the best cables (HDMI if you have it plugged in, PS3 video-out if you don't) you have plugged into it until it displays an image that allows you to respond to it. At that point you can tell it what resolutions your display is capable of showing and set it up properly. This is covered in your PS3 manual.
I doubt it'll work through an HDMI-to-DVI adapter because of HDCP.
And this question has nothing whatsoever to do with Gran Turismo 6.
Yes, you have to be sure both the cable (and any adaptors used either end) and the monitor support HDCP otherwise it won't work. I used to have my PS3 plugged in to a Samsung 226BW monitor (which was made after HDCP became a 'thing') using an HDMI to DVI cable.