There are a lot of texts which talks about coordinate and components of a vector and Coordinate system vs basis of vector space, which are all using the terms interchangeably. Nowhere have I found these terms being explained so as to properly differentiate them.
What are coordinates of a vector and components of a vector? I know that the in contravariant vectors, the coordinates of a vector and components of the vector vary the same way and that in a covariant vector they they change oppositely . Also, I have read that when a basis is chosen for the vector space the coordinates of the space (so coordinate system?) are set but that the components of the vector don't need a basis,i.e. components are part of the vector itself and so don't need a basis( I don't understand how components don't need any basis). As per my understanding coordinates of a vector are the coefficients which along with basis vector (i.e. a linear combination of them) give any vector in the vector space(span the vector space). So the coefficients i.e the coordinates of the vector are given relative to the basis. But coordinate of a vector space is (is this coordinate system?) is a function (which is set on choosing the basis set), which maps a vector $V$ (in vector space) to its coordinates (in that basis set). $$x^i[f]v=v^i[f]$$ Where $x^i$ is the coordinate function(does a coordinate system for a vector space mean this coordinate function?) in the basis $f$ and $v$ is the vector in a finite-dimensional vector space and $v^i$ is the coordinates of that vector in basis $f$. So is this the function $x^i[f]$ (that is called the coordinate system of vector space) that vary similar to coordinate(components) of the vector in a contravariant vector? I have also read that coordinates are the function that map region of the euclidian space to points from a manifold but let's not use that explanation. Is the components of the covariant vector mean that they are the dot product of the vector and basis vector,i.e the projection of the vector along each of the basis vectors( if this is so then components of vector do need the basis for their definition too) and that is why they c0-vary with changes to basis in covariant vector?
Also, what is the difference between the coordinate system of a vector space and basis vector for a vector space or more precisely does a change of coordinate system mean the same thing as the change of basis vector set?. I know if the change of basis set is given by a linear map then it is same a change of coordinate system too but is it the only instance where they both are considered similar as the change of coordinate system to a cylindrical or spherical coordinate system won't be considered as a change of basis. What are the definitions of change of coordinate system and of change of basis and is the change of coordinate matrix the same as the change of basis matrix?