How to write a vector in a rectilinear co-ordinate system I understand that the covariant and contravariant components of a vector are the same in Cartesian co-ordinate system, but they are generally different in some curvilinear co-ordinate system. When writing a vector, we write the components multiplied by the corresponding base vector.                                   
What I don't understand is, when we write the covariant components of a vector we put the corresponding covariant base vector with it and not the contravariant, why is that? 
 A: Vector spaces have an axiomatic description; here, a basis can be chosen and with respect to this basis any vector can be expressed via a set of components; in this description there is no notion of a covariant & contravariant vector.
However, once we introduce an inner product:

$g: U \times V  \rightarrow R $

a distinction does arise in a symmetric way because of the way we treat bases and change of bases here; suppose we have a basis $e^i$ of the vector space $U$ and a basis $f_j$ of the vector space $V$. We say that they are dual when

$g(e^i, f_j) =\delta^i_j$

Now suppose we transform the basis of $U$, then we can ask how does the basis of $V$ change in order to keep this equation invariant? Suppose, as it must be, the former transformation is effected by a linear matrix $T$ then the latter changes by $T^{-1}$; it's for this reason that we call the vector space $U$ covariant, and $V$ contravariant. 
This is touched on, but not fully explained, in Diracs book on GR; and expanded upon in Weinbergs book on the same. It's common nomenclature in physics but not so much in mathematics, here a different notion of covariance and contravariance is given that is coordinate free and doesn't rely on bases. 
Given a map between manifolds 

$f: M \rightarrow N$ 

Then we have the tangent functor $T$ 

$Tf: TM \rightarrow TN$

and the cotangent functor $T^*$

$T^*f : T^*N \rightarrow T^*M$

Notice that the arrow for $Tf$ goes in the same direction as $f$, for this reason we say that $T$ is a covariant functor, or just a functor; and notice, that the arrow for $T^*f$ goes in the opposite direction to $f$, and for this reason we call it a contravariant functor, or just a cofunctor.
So the tangent functor is covariant and the cotangent functor is contravariant; this notion of covariance & contravariance is pervasive in mathematics and different from the physical one already described (though related).
