I General Linear Group GL(n) on Vector Spaces and canonical pairing invariance

[jv07cs]

Does anyone have a reference that explains how the general linear group GL(n) acts on vector spaces and dual spaces? Furthermore, I would like to understand why the canonical pairing $\langle\cdot, \cdot\rangle: V \times V^* \to \mathbb{F}$, $(v,\alpha) \mapsto \langle\alpha,v \rangle := \alpha(v)$, is GL(n) invariant.

 

[mathwonk]

In my opinion,

GL(n) acts, not on V, but on k^n, so it acts on the n dimensional space V only after choosing an ordered basis for V, (and thus also an ordered dual basis for V*).

Then if a is a covector in V*, represented by a row vector, and v is a vector in V, represented by a column vector, and if M is an invertible nxn matrix, then M takes a to aM and takes v to Mv, hence the fact that the matrix product aMv is associative, i.e. (aM)v = a(Mv), expresses the GL(n) - invariance of the pairing taking <v,a> in VxV*, to the dot product (a.v).

The group that acts on V, is called GL(V), and is by definition the group of linear automorphisms of V, hence it acts on V by definition, exactly as above, i.e. M in GL(V) takes v to M(v), and takes a to the composition aoM. Hence as above, (aoM)(v) = a(M(v)), shows the invariance, which here is actually a definition.