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?