Well that all depends. A speaker may well do but it will still require an amplifier. And car amplifiers wont work on mains, Unless you work on it and install a VERY large cap as it will either cause amp shutdown or can pop or trip a 10A fuse.
Sure is possible, but unless you get a good one, and that basically rules out 90% of them then you will just get the same crappy resonant one note performance they give you when used in a car.
The beauty of active subs which is what most home ones are is they can correct the response of the unit before the amplifier, and in lots of them add feedback making them a servo design. Getting a cheap nasty $80 12" and putting it in a $40 box from repco will just get you something loud and rattly that really only belongs in a car if you like to sit really low in your seat and spike your hair up with excessive amounts of gel and drive you lowered civic around trying to pick up slappahs...
You will also need an amp to drive the sub. These look cheap enough... http://www.globalmediapro.com/do/product/1818 - I haven't got one yet, but a mate has an amp that looks very similar and it can get good power levels on his dual 18"ers in his home theater.
When you compare the cost of a good sub, building a box for it, and the amp, then a ready made solution from velodyne or similar starts to look like good value.
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