I recently taught a junior camp where it surprised me how many kids were taught to swing the club down the line. This misconception about the golf swing led to many other faults to occur.
It is important to remember that the swing is a circle. There are no lines in a circle. The only thing that goes down the line is the golf ball after it is hit. The club only intersects with the target line for a split second after the ball is struck.
The club is moved by the body back around and to the inside on the back swing and the follows the body back around to the inside on the forward swing. Imagine a hula hoop laying on its side at a 45 degree angle that would be the look of the swing.