What is the difference between vector components and its coordinates? Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is meaningless to talk about coordinates when basis is unspecified. But don't components depend on basis? Why can you talk about components regardless of the basis? Do they just demonstrate their domination by being too picky and making the difference between the things, which are the same, and, thus, causing confusion? Later they mock my definition of vector space, where I say that it is a collection of vectors. They say that it is wrong since vector space an Abelian group whose elements can be scaled. But, I see no difference between these definitions.
edit If this answer and Wikipedia are correct then $\psi_i = \langle \psi|i\rangle$ in 
$$\vec \psi = \begin{bmatrix}|1\rangle |2\rangle \cdots |n\rangle\end{bmatrix} \begin{bmatrix} \psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \end{bmatrix}$$
must be coordinates in the basis $\begin{bmatrix}|1\rangle |2\rangle \cdots |n\rangle\end{bmatrix}$, which I believe is different from the standard basis. Why do quantum mechanics say that $\psi_i$ are components then rather than coordinates? Is it incorrect?
Miami Operators and Matrices also say 

Now you see where the defining equation for operator components comes
  in. Eq. (7.7) is
  $$\sum_k u_k \vec e_k = \sum_i v_i \sum_k f_{ki} \vec e_k$$

Actually, I was motivated to ask this question when tried to represent these equations, which involve integration, in vector form and do not understand why everybody calls the coefficients $u_k$ in $\sum u_k \vec e_k$ components rather than coordinates. It seems to me that matematicians use these terms interchangably and troll the noobs with immaterial difference.
PS Dec 2013
I see here 

that components, $a_i \psi_i$ are contrasted with coefficients $a_i$. Are components just another names for coordinates or we have a 3rd type of object? 

Components are used to describe general vectors in a given vector space within a basis in terms of linear combinations. Components of a given vector also are coordinates in vector spaces, because univocally determine the position. But talking in more general spaces (like manifolds) I think that nobody says that a given position has certain "components" instead of coordinates.  
