Pure Mathematical Approach? What is it? I am reading a textbook Linear Algebra: A Pure Mathematical Approach by Harvey E. Rose. I am finding it a bit difficult, because the author doesn't always explain everything so well. And, when I went to a different textbook to clarify permutation groups I returned to my book to find that the author did it differently  he does permutation from left to right.
All of this has me asking, what is the pure mathematical approach? Is this like a different branch of mathematics?
And why is everything left to right? 
The normal $f\circ g(x)=f(g(x))$  is now $x(g\circ f)=x(gf)=f(g(x))$. 
 A: BIS HD gave a good answer to what it means to take a pure mathematical approach at a subject. To add to BIS HD's answer, I talked to a professor today and he explained (essentially) that the study of permutations came about before functions were defined in the way that they are today. And, originally in the study of permutation it was natural to to write a permutation, composed of two seperate permutations (say $\sigma$ & $\tau$), on a set $A$ as 
$$A\tau\sigma$$ 
Meaning morph $A$ into $\tau (A)$ and then morph $\tau (A)$ into $\sigma (\tau (A))$. Later when the study of functions became more developed the common notation  $f(x)$ came along which wrote the object to be changed (or morphed) $x$ on the other side of the transformer $f$, in this respect it became common to write things like $(g\circ f)(x)$ which can be read "$g$ follows $f$". This new way of writing compositions, eventually made its way to other part of mathematics including the study of permutation groups. Today, it is not uncommon for an author to write the same permutation above as $\sigma\tau(A)$, opposed to $A\tau\sigma$. 
This new style of notation is some what unnatural (if you were not taught this at first), because we read and write from left to right (at least in English). So, some Algebraist prefer use the original notation $A\tau\sigma$,  and in the same vein the familiar composition $(g\circ f)(x)=g(f(x))$ is written as $x(fg)$ which naturally says  let $x$ be morphed be $f$ and then let then morph the resulting object by $g$.
A: Some authors like to denote the action of a function on an element (or of a group on an element) as a left-action, some (as H. Rose does) prefer the right-action. This is in fact a different notation, that hasn't any effects but in my opinion readability. 
Linear algebra can be approached by applied mathematics because there are a lot of algorithms that you can employ e.g. Gaussian elimination. Those algorithms are used for example in numerical mathematics, a branch of applied mathematics. An approach through pure mathematics means to be as formal as possible about notation and to focus on the conceptual parts of linear algebra, e.g. vector space homomorphisms, eigenvalues, etc. Hope I could help you.
