Basis of a vector space

A useful observation is that for any vector space $V$ of dimension $n$ , giving a particular basis $B$ is the same that choosing a particular isomorphism

b : R^{n} ⟶ V

In this context, the coordinates or components of $v \in V$ are given by $b^{- 1} (v)$ . Moreover, a basis change is identifiable with an element $P \in G L (n, R)$ in the sense that $b \circ P$ is a new isomorphism of $R^{n}$ into $V$ . That is, if we have two basis $b_{1}, b_{2} : R^{n} \mapsto V$ , the basis change is a $P : R^{n} \mapsto R^{n}$ such that

b_{2} = b_{1} \circ P

On the other hand, any transformation $T \in G L (V)$ can be seen as a change of basis. If we fix a basis we have

b_{1} : R^{n} ⟼ V \overset{T}{⟼} V

We can define $\tilde{T} = b_{1}^{- 1} T b_{1}$ (basis change) and $b_{2} = T b_{1}$ (new basis), so that we get

b_{2} = b_{1} \circ \tilde{T}

If we denote by $B_{V}$ the set of all the basis, what we have presented is a right group action of the general linear group $G L (n)$ in $B_{V}$ .

This approach is important in principal bundle#Alternative approach and homogeneous space#Intuitive approach. It is generalized in general covariance and contravariance.