This sounds like either the source has the line (not likely but your post has me second guessing it could be the source and not the TV.) My test would be to see if the line is in the menus and setup areas too.
Frankly the cheap exit is if your set was from Costco (2 years!) and some credit cards. As this is rarely user serviceable, buy the extended warranty and then make the call for repair.
Bob
I have developed a problem with my samsung tv. I purchased the tv new in November or December of 2013. A few weeks ago i noticed the tv had a single thing black line going all they way across my tv. The line was still there after trying multiple imputs. When i do the picture test, the lines are not visible. They come back as soon as the test is done. It is very frustrating to pay as much as i did to have it start having problems when it is just over a year old. Anyone know what the problem might me?

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic