What is linear, numerically and geometrically speaking? For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear .
Intuitively anything that looks like a straight line is interpreted as linear, like something in the form $f(x) = mx + q$ or any other function that maps $R \rightarrow R$ resulting in a graph that looks like a line.
Last time I had the chance to talk to a physicist, the guy made a "meh, not really" expression about this, meaning that I had the feeling that this is "true" at some conditions but it's not a rigorous and universally accepted definition.
So, since we are talking about this, could you define the concept of linear from the geometric and numeric point-of-view ?
So maybe I can really grasp what "linear" means everytime this adjective appears in the names of math topics like linear algebra, linear programming, and so on.
 A: We normally say that a function is linear, and normally in the following (slightly informal) sense: Let's suppose that a function $f$ is mapping elements from $A$ to $B$ and both $A$ and $B$ have a way of both adding elements together and also multiplying elements by scalars (what I really want to say is that $A$ and $B$ are vector spaces but don't worry if you don't know what a vector space is - just think of it is a place that you can add elements together and multiply by scalars).
Now, we say that $f\colon A\to B$ is linear if for any points $x,y$ in $A$ we have $f(x+y)=f(x)+f(y)$ and for any real number $r$ we have $f(r\cdot x)=r\cdot f(x)$.
In the special case that $A$ and $B$ are both $\mathbb{R}$, the graph of a linear function is in fact a straight line in the plane $\mathbb{R}\times\mathbb{R}$ and so at least in this sense it satisfies your intution. In order to get a better geometric intuition, you really need to take a course/read a book on linear algebra - generally a first year module in any good mathematics degree at the university level.
A: A linear function is any function that satisfies, in general, $f(ax+by) = af(x)+bf(y)$. This function could be one dimensional or not.
Geometrically, linear functions tend to look like lines or their higher dimensional equivalents. For instance, a 2-D linear function looks like a flat plane, 3-D looks like a cube, etc...
However, not all things that look like lines are linear. Note that $f(x) = ax+b$ does not satisfy $f(x+y) = f(x)+f(y)$. If we wish to be very precise, we call this function affine. In essence, an affine function is a linear function plus a constant shift, which is manifest by the the existence of the constant $b$.
The distinction is really one of our coordinate system, however. In such a case, we can choose a slightly different coordinate system by taking the map $x \mapsto x-\frac{b}{a}$. Then, $ax+b \mapsto a(x-\frac{b}{a})+b = ax$. Now, in this shifted coordinate system, we do have a linear function, and we can treat it as such for the purposes of our analysis. We can go back and add $b/a$ to our results later, if we want. (Sometimes).
So, for that reason, we tend to call all degree-1 polynomials "linear", even if they aren't strictly linear.
A: The concept of linearity can be explored easily enough for real functions.
Formally, a function $f:\mathbb R \rightarrow \mathbb R$ is linear if $f(x+y)=f(x)+f(y)$.
Note that any function of the form $f(x) = ax$ has this property since
$$
f(x+y)= a(x+y) = ax+ay = f(x)+f(y).
$$
In fact, it can be proved that the only continuous functions satisfying $f(x+y)=f(x)+f(y)$ are the functions of the form $f(x)=ax$.  Note that $f(x)=ax+b$ doesn't cut it, unless $b=0$, because
$$
f(x+y)= a(x+y)+b \neq (ax+b) + (ay+b) = f(x)+f(y).
$$
Intuitively, we might say that a function is linear if its output is proportional to its input.  If we double in the input, we double the output.  If we triple the input, we triple the output.  This is again a consequence of the defining property:
$$
f(2x) = f(x+x) = f(x)+f(x) = 2f(x).
$$
It's also easily derived from the form $f(x)=ax$.
More generally, a function $f:{\mathbb R}^n \rightarrow {\mathbb R}^m$ is linear if $f(x+y)=f(x)+f(y)$ for all $x,y\in{\mathbb R}^m$ and $f(\lambda x)=\lambda f(x)$ for all $x\in{\mathbb R}^m$ and $\lambda \in \mathbb R$. The symbol $\lambda$ is sometimes called a scalar and isn't really necessary when $m=n=1$, as above.  In this more general case, it turns out that any such function can be represented in the form 
$f(x) = Ax$, where $A$ is an $m\times n$ matrix.
Even more generally, we might discuss linear functions $f:U\rightarrow V$, where $U$ and $V$ are vector spaces.  Things get a bit more abstract at this point.  The matrix representation might no longer be good as the spaces might be infinite dimensional.  The same ideas still apply, however.  We still need $f(x+y)=f(x)+f(y)$ and $f(\lambda x)=\lambda f(x)$.  Differentiation is an example of such an operation, since
$$\frac{d}{dx} (g(x) + h(x)) = g'(x)+h'(x).$$
Care needs to be taken with the domain; this operation maps the set of continuously differentiable function to the set of continuous ones for example.
