Morphism between matrices and linear equations I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to handle linear equations easily. However they never establish the connection between both objects and never explain why it is possible to work with matrices in substitution of linear equations. 
For some classmates this is irrelevant because they say that I just complicate my life with such questions. But it is important and think that the treatment given in such books is either very informal so beginners like me can understand the concepts or maybe is too simple that I'm missing something. I have read that two objects are generally treated as being the same, of course under certain properties, if there is a connection between them in terms of a one-to-one correspondence (something called morphism, isomorphism, monomorphism, etc). 
So, how would you establish the bijection between linear equations and matrices considering elementary operations? 
 A: The general form of row $i, \, 1 \leq i \leq m$, of a system of $m$ linear equations in $n$ variables $x_j$ can be written as
$$\sum_{j=1}^n a_{i,j} x_j = b_i,$$
$a_{i,j}, b_i \in \mathbb{R}.$  
The general form of row $i$ of an $m \times n$ matrix $A$ can be written as
$$(a_{i,1}, \dots, a_{i,n});$$
the general form of a vector $b \in \mathbb{R^m}$ as
$${(b_1, \cdots, b_m)}^t.$$
You are looking at the matrix equation of form
$$Ax = b,$$
for the same values $a_{i,j}, b_i$ as in the system of equations.  
Let $E$ be the space of such equations for $b_i == 0$ (as we only map to the matrix, the LHS of the matrix equation), described by its entries $a_{i,j}$, and $M^{m,n}$ be defined in the usual way. So there is a natural bijection $\phi$ between the two, mapping
$$\phi: E \rightarrow M^{m,n}, \quad \{a_{1,1}, \dots, a_{1,n}; a_{2,1}, \cdots , a_{m,n} \} \mapsto
A = (a_{i,j}).$$ 
It's probably obvious that this map is a bijection (also an isomorphism) between the two spaces. Is this what you were looking for? Or was it that applying elementary matrices to a system represented by a matrix equation is the same as performing row-operations on the system, and that these operations do not change the solution space? Showing that is not hard, but would take a fairly longish series of lemmata. 
