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
$$ b:\mathbb{R}^n \longrightarrow V $$In this context, the coordinates of $v\in V$ are given by $b^{-1}(v)$. Moreover, a base change is identifiable with an element $P\in GL(n,\mathbb{R})$ in the sense that $b \circ P$ is a new isomorphism of $\mathbb{R}^n$ into $V$. That is, if we have two basis $b_1, b_2:\mathbb{R}^n\mapsto V$, the basis change is a $P:\mathbb{R}^n\mapsto \mathbb{R}^n$ such that
$$ b_2=b_1\circ P $$On the other hand, any transformation $T\in GL(V)$ can be seen as a change of basis. If we fix a basis we have
$$ b_1:\mathbb{R}^n\longmapsto V \stackrel{T}{\longmapsto} V $$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
$$ b_2=b_1\circ \tilde{T} $$I 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 $GL(n)$ 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.
See also covariance and contravariance in linear algebra.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: