An example of a linear function; We must show closure under scalar multiplication, and vector addition.
Define F by: F(X) = (X,Y)
F(X1+X2) = (X1+X2,Y) = (X1,Y) + (X2,Y)
= F(X1) + F(X2)
F(aX) = (aX,Y) = a (X,Y) = a F(X)
So F is linear, other examples include ordered tuples, and directed line segments.