I'm planning to buy a new graphics/video card/adapter AGP 8x or 4x, 0.8 V or 1.5 V for my SUSE 9.3 Pro machine. The resolution of my VGA computer screen is 1280x1024. My TV is a Sony Trinitron Widescreen KV-16WT1 16 inch. I don't know the resolution of it. I don't want to change the resolution of my computer screen and I want a cloned image of my computer screen on the TV. Since my PC screen is 4:3 and my TV is 16:9 I can accept that the image gets wider and flatter on my TV. The TV system is PAL-B/G. I have a graphics card with TV-OUT right now but it's really old and I don't think it can handle cloning a PC screen with a resolution of 1280x1024 (see here "The TV-out has a max of either 1024x768 or 800x600 depending on...") (This is my present graphics card NVidia Riva TNT2 32MB AGP Video Card with TV-Out CT5823)

Which graphics card should I buy to achieve a maximum likelihood of getting the TV-OUT work the way I want? (one resolution on the PC screen and another on the TV (since it's not 1280x1024 px))

For the best Linux support out there at this time, I'd stick with nVidia. They really do have good Linux support. Not sure on a particular model, but most of the newer models should be more than satisfactory. I've used a GeForce 2 MX, and a GeForce Ti 4200 and received excellent results. For more detailed reviews, you should check out mythtv.org, as this is exactly one of their design criteria (they develop an open source personal video recorder software package, kind of like Tivo on major steroids). They will have a list of video cards with svideo, RCA, and LCD output capabilities that work well with Linux.

Get an ATI card with two VGA outputs for optimal TV out. ATI can support RGB output via the scart port on your TV. It takes a bit of tweaking with modelines though but it can be done (I've done it), and it's definitely worth it in terms of picture quality. At least if you want to use the computer for watching movies.