I'm a little bit confused about how coordinate systems work once we have chosen a basis for a vector space. Let's take R^2 for example. It is known that if we write a vector in R^2 numerically, it must always be with respect to some basis. So the vector [1, 2] represents the point (1, 2) in the xy-plane if we take our basis vectors to be the standard ones. However, I am a bit confused by this. If we let i and j be the basis vectors, the the coordinate vector [1, 2] is more fundamentally represented by 1i + 2j. However, what then, do i and j represent? One might say that i is [1, 0] and j is [0, 1]. However, this definition seems circular, because we are then using the standard basis to define the standard basis vectors... So how are i and j defined, without referencing the standard basis to define them? The same question goes for any other basis vectors we may find. How can we define these basis vectors without having to reference the standard basis?