A linear transformation is one where only linear operations are performed on the old x and y values to get the new ones. In practice, this means you can't multiply any two variables; x cannot become xy, y cannot become x^2.
they also have the other property that if operated on the origin (0,0), it will not move.
I was confused by 4D as well. if it helps, think of it this way; for a transformation to be linear, you can only add x and y terms and you can only multiply by constants. so any (x+n) where n is a constant like 1, 3, 5 etc. is not linear because if operated on the origin, it would move the origin.
heads up for when you do matrix transformations; think about the matrix being operated on the identity matrix, and think of the identity matrix as two coordinates of (1,0) and (0,1). if this last bit isn't clear, don't worry just yet, just make sure you understand what a linear transformation is first.