coordinate isomorphism

Fix a basis e_1, ..., e_n of an n-dimensional space V over F. Every vector v then has unique coordinates (c_1, ..., c_n) given by v = c_1*e_1 + ... + c_n*e_n. The coordinate isomorphism is the map sending v to that tuple, [v] in F^n. It is the formal statement that 'writing a vector in a basis' is itself a vector space isomorphism V -> F^n — bijective, linear, structure-preserving.

The slogan to carry away is: choosing a basis IS choosing an isomorphism to F^n, and conversely. The two acts are literally the same data. This is why coordinates feel so natural in Vol I — you were tacitly working through a coordinate isomorphism the whole time, turning abstract vectors into columns you could compute with. The isomorphism is the bridge between the coordinate-free world and the arithmetic of F^n.

The essential caveat: this isomorphism is NOT canonical. There is no preferred basis of an abstract space, so there is no preferred way to coordinatize it — a different basis gives a different, equally valid coordinate isomorphism. The vector v is one fixed object; its coordinate tuple depends entirely on the basis you chose. Change of basis is precisely the change-of-variables formula relating two such isomorphisms via an invertible matrix.

Keeping the non-canonicity in view is what separates fluent from fragile understanding. A coordinate-free statement (this operator is invertible, this subspace is invariant) is true regardless of basis; a coordinate statement (this matrix is upper triangular) may hold in one basis and fail in another. Good linear algebra constantly translates between the two views — using coordinates to compute, but trusting only the basis-independent conclusions.