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

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

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

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

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.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es