Properties of Linear Transformations? 
A linear transformation, $$T: \Bbb{R}^m \rightarrow \Bbb{R}^n$$ is a function that has the following properties.
$$T(\text {u} + \text v) = T(\text u) + T(\text v)$$
  $$T(\text{kv}) = \text kT(\text v)$$
where u and v are vectors and k is a scalar.

Why do linear transformations have to have those properties? Normal functions like $f(x) = x^2$ don't have those properties, so why do linear transformations have to have them?
 A: Maps between algebraic objects are best when they respect the structure of the objects.  In the case of vector spaces, the structure includes two operations: vector addition and scalar multiplication.  The best type of map to define between vector spaces is thus a map that respects these two operations.  This is exactly what we require of a linear transformation.
To be more specific, the definition of a (real) vector space $V$ begins with a set of elements, called vectors.  Just about any collection of objects is a set, so as it stands, we haven't required much of something to be called a vector space.  In order to make the object (the vector space) a bit more focused, we should give it some more structure, or in other words, we should require that the vector space satisfy certain conditions.
The first of these conditions is that if we are given two elements in $V$, represented by the symbols $u$ and $v$, we have a way of "adding" the two vectors to obtain another vector, which we denote by the symbol "$u+v$."  This operation, called vector addition, gives our set $V$ more structure than just a normal set.
The second condition is that given any real number $k$ and any vector $v$, we can "multiply" $v$ by $k$ to obtain a new vector, which we denote $kv$.  This operation, called scalar multiplication, gives our set $V$ even more structure.
Now, we don't allow vector addition and scalar multiplication to be completely arbitrary, but we require that they interact with each other in a nice way.  For example, we would like the two elements represented by the symbols $k(u+v)$ and $ku+kv$ to actually be the same element for all possible choices of real numbers $k$ and vectors $u$ and $v$.  This is the distributive rule which gives our vector space even more structure.  The rules that our two operations must satisfy are called axioms.
In summary, we have constructed an algebraic object $V$, by starting with an arbitrary set, and then requiring that this set have certain operations satisfying certain rules.  These operations and rules are the structure given to $V$.
Now a map, $T$, between two vector spaces $V$ and $W$ is, at its core, just a map of sets.  That is, we have a rule that associates to each $v$ in $V$ exactly one $w$ in $W$.  We write $T(v)=w$ to represent this association.
Now, just as we required a certain level of structure on the objects $V$ and $W$ to be called vector spaces, we should also require a certain level of structure on our map $T$ if we would like to consider it a map between vector spaces.  That is, we don't want to allow any arbitrary map of sets.  So what should we require of $T$?  Well, it is fairly natural to require that $T$ preserves the structures given to $V$ and $W$.  This means that if $T$ associates $u$ to $x$, and $v$ to $w$, then it should associate $u+v$ to $x+w$.  In other words, if $T(u)=x$ and $T(v)=w$, then $T(u+v)=x+w=T(u)+T(v)$.  Mathematicians say that $T$ "respects the operation of vector addition."
If $T$ also respects scalar multiplication, as you have written down, then in some sense $T$ respects all of the structure of a vector space, and so we say that $T$ is a map between vector spaces $V$ and $W$.  We call such maps linear transformations.
In general, this process is a theme in algebra.  Start with a set, and give the set structure with operations and other rules.  What we get is an algebraic object.  Then, to define a map between algebraic objects, define a map of sets that in a natural sense respects the operations and rules (i.e., the structure) we've place on the sets.
