What does being "Linear" mean for a transformation and a function intuitively/graphically? I was wondering what is the geometric meaning or intuition behind a transformation and function(separately)being linear.
An example(or graph) illustrating the characteristics of a linear function/map would be much appreciated.
Thanks in advance.  
EDIT: Also, I have read that if we "zoom in" the graphs of some functions, we see that they "become linear around the point of magnification"(you can find it in Callahan's Advanced Calculus". What does that mean?

If you have a linear transformation on a space $X$ then the image is a subspace of the space $X$. Geometrically that means that the image of the transformation is a flat that contains the origin in the space.
Examples: 
The easiest example is the 0-transformation. 
Let $X = \mathbb R ^n$ and $f: X \to X, v \mapsto 0$ then $Im(f) = \{0\}$ and thus just a point.
The next easiest example is the identity:
Let $X = \mathbb R ^n$ and $f: X \to X, v \mapsto v$ then $Im(f) = X$ and thus the whole space.
A more complex example is:
Let $X = \mathbb R ^n$ and $f: X \to X, v \mapsto (v_1, 0 , \dots, 0)$ then $Im(f) = \{(v_1, 0 , \dots, 0) \mid v_1 \in \mathbb R\}$ and thus a line.
Hope that helps you :)
