Linearity In Linear Algebra I am learning linear algebra for few months now and I came to the following notion.
Due to the definition of field:
$\sum_{i=1}^{n} \alpha(a_i+b_i)=\alpha\sum_{i=1}^{n}  a_i+\alpha\sum_{i=1}^{n} b_i$
Due to the definition of vector space if $a,b\in V $ and $\alpha\in F$ then $\alpha(a+b)\in V$, and $T:V\rightarrow U$ is a linear transformation then:
$\alpha T(v_1+v_2)=T(\alpha v_1)+T(\alpha v_2)$
All of the above says that in Linear Algebra all of the linear operations preserve linear combinations? am I missing something? 
 A: Right...you've hit upon something that your book should have made explicit. I'd have written something like this:
We're looking at fields (like $\mathbb R$) and coordinate vectors over those fields (like $\mathbb R^3$), and when we define transformations, we ask that these transformations preserve field properties. Why? Because structure-preserving maps have proved, over the years, to be ones about which we can say meaningful things. For instance, a little later in the book, we'll talk about "dimension", and we'll show that if $T$ is a linear transformation from $V$ to $W$, then (1) $T(V)$ is a subspace of $W$ (essentially, a vector space that's a subset of $W$), and (2) the dimension of $T(V)$ is no greater than the dimension of $V$. So linear maps preserve-or-reduce dimension. If we look at nonlinear maps, it's quite possible for $T(V)$ to have a dimension greater than that of $V$. We'll see this pattern over and over: linearity --- i.e., preserving the algebraic structure --- is exactly the secret sauce needed to make some statement true. This is just the first instance of many. 
A: This is not just the case in linear Algebra, as a linear operator is by definition an operator that preserves linear combinations. (correct me if I'm missing some field of mathematics, where the term "linear operator" has a different meaning)
