# general linear group

Let \(V\) be a vector space over a field \(F\). Then \(GL(V)\) is the group of all automorphisms of \(V\). That is, \(GL(V)\) is:

- the set of all mappings \(V\rightarrow V\)
- function composition as the group operation

When the vector space is finite dimensional it turns out that \(GL(V)\) is isomorphic to \(GL(n, F)\), the group of \(n\times n\) invertible matrices with entries from \(F\), with matrix multiplication as the group operation.

## 1. bases

If we choose a basis for \(V\), then we have an isomorphism from \(GL(V)\) to \(GL(n,F)\) which is the set of \(n\times n\) invertible matrices with entries from a field \(F\).