What's the relation between basis for a vector space and coordinate systems?

Question

I know what's a basis for some vector space $V$: a set of objects from that space that span the whole space.

We can change between basis by using the change of basis matrix. Basically this matrix transforms a vector representation with respect to a basis to another representation with respect to a new basis.

I'm now wondering what's the relation between a basis of a vector space and a coordinate system for that same vector space?

Answer 1

In a (finite dimensional) vector space (over a field), one cannot talk about coordinates without referring to a basis.

Given an ordered basis, you have a corresponding "coordinates system" by definition:

Given a basis of a vector space $V$, every element of $V$ can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components.