The first postulate of quantum mechanics is about states being elements of a vector space. In mathematical literature, a systematic procedure for representing the vectors as more familiar objects is known  as making a choice of, and working with, a representation.

Physically, introducing  a representation amounts to a process that begins with associating  vectors as collection of all probability amplitudes of simultaneously measurable quantities. This is explained with the help of an example.

This approach does not critically require knowledge of vector spaces. Most of the  mathematical details concerning  vector spaces, operators and representations are bypassed. However, for a complete understanding and for a clear overview, the standard process of introducing representation in vector spaces and a knowledge of transformation theory is recommended.

Who may find this article useful?
This is the article for you if you reached one of the following states, or a "quantum superposition'' thereof.

• Your teacher started teaching QM and has just finished postulates. He/she  tell your class that he/she  needs to go through choice of an orthonormal basis, introducing representations, and change of representation and a lot of 'esoteric' mathematical stuff, before coming to more down to earth discussions involving free particle, particle in a box and so on. But you are getting impatient to start computing, or getting tired of mathematics.
•  You had a course in Quantum Mechanics. You were never taught about  postulates and vector spaces. You have just managed to learn postulates yourself. Every one tells you that the states are represented by a vector in a complex vector space. But no one, a book or a person, tells you what these vectors are? They all talk abstract vector space language and assure you that physical connections will come at the end and there will light at the end of the tunnel. You have searched internet and library books and have not found a clear cut answer.  But you are getting impatient to to see the physical connection of abstract state vectors that are used.
•  You have tried learning postulates yourself and you have gone through all mathematics involving vector spaces, orthonormal bases, representations and change of basis and have understood all. But you still long for a more direct and a physical approach to the questions like

(a) Given a system, what is the vector space and what realization(s) of vectors can one have?

(b) For a given system why we can have so many different looking  descriptions? Momentum space wave functions, Schrodinger wave function, and so on ..?

(c) Why the hell we need two component wave functions for electrons when you  need to talk about spin

• You are a teacher you have patiently taught postulates and later connected them with coordinate and momentum representations but your class keeps asking for a physical approach.
• You are a teacher of a course where some QM is needed as prerequisite,
• But you do not have lecture time to develop all mathematics, but still you want to start from postulates and take the class quickly to the Schrödinger wave mechanics or quantum mechanics of spin systems.

What do I need to get started?
I will give a small list of what all you generally need to know to get started. Any specific thing that is needed will be given later.

1.   Elementary ideas about vector spaces.
2.   Indeterminacy in quantum mechanics and probabilistic nature of quantum theory.
3.   Simultaneous measurements and uncertainty principle.
4.   Postulates of quantum mechanics.

For quantum systems there is uncertainty relation between position and momentum. You may be wondering whether there are other  pairs for which there is some kind of uncertainty relation? The answer is that if there is an uncertainty relation for dynamical variables $A,B$ if the corresponding operators $\hat{A},\hat{B}$ do not commute.