Historical reason to define a matrix vector product the way it is What is the reason why we defined a matrix vector product (a transformation) this way:

$$\begin{pmatrix} a_1 & a_2 \\  a_3 & a_4  \\  \end{pmatrix}\cdot \begin{pmatrix} b_1\\  b_2\\ \end{pmatrix} = \begin{pmatrix} a_1\cdot b_1 + a_2\cdot b_2 \\  a_3\cdot b_1 + a_4\cdot b_2\\ \end{pmatrix}.$$

I know that when we want to represent a transformation, we care about the values that we're gonna multiply the vector's pieces, but why we use like a linear combination mechanism to multiply the vector's pieces?
I know that this may be a good reason:


*

*We can create new vectors with more or less dimensions than the original vector, just by having one, two, or more lines in the matrix (os less lines).


PS: I can't just say that this is derived from the matrix matrix product, because this type of product is derived from the matrix vector product definition, so any 'proof' or reason that uses matrix matrix product definition will be circular.
 A: In spite of your PS, I believe that matrix matrix product is the reason, since every $n\times m$ matrix $A$  can be seen as linear transformation $\mathcal{A}: \mathbb{R}^m \to \mathbb{R}^n$. If you choose basis $\{e_1, \;... \;,e_n \}$ for $\mathbb{R}^n$ and basis $\{f_1, \;... \;,f_n \}$ for $\mathbb{R}^m$, then $A$, $A_{ij} = a_{ij}$, gives you information about $\mathcal{A}$-images of $f_j$-s: 
$$\mathcal{A}f_j = \sum_{i = 1}^n a_{ij}e_i\text{.}$$
If you have two such linear transformations $\mathcal{B}$ and $\mathcal{C}$ (with corresponding matrices $B$ and $C$) for which composition $\mathcal{B}\circ \mathcal{C}$ can be computed, the matrix matrix product of $B$ and $C$ gives you matrix $BC$, which corresponds to linear transformation $\mathcal{B}\circ \mathcal{C}$. 
I agree that the last fact is not intuitive and my personal opinion is before the matrix matrix product was defined, they had asked themselves
Which matrix corresponds to composition of two linear transformations? 
Let be  $\mathcal{C}:X\to Y$ and $\mathcal{B}:Y\to Z$. Choosing basis for $X$ ($e_i$,), $Y$ ($f_i$,) and $Z$ ($g_i$,), it can be directly computed that $i$-th
component of $(\mathcal{B}\circ \mathcal{C})e_j$ equals to
$$\sum_k b_{ik}c_{kj}$$
so the matrix matrix product should be defined as it is nowadays.
