# Kernel of GL(n,F) acting on F^n

## Homework Statement

Suppose GL(n,F) acts on F^n in the usual way. Consider the induced action on the set of all k-dimensional subspaces of F^n. What's the kernel of this action? Is it faithful

## The Attempt at a Solution

Well, I anticipate that the kernel of this action consists of scalar matrices, that is scalar multiples of the identity matrix. The question is how to prove that if g in GL(n,F) is a not a scalar matrix then we can always construct a k-dimensional subspace not fixed by g.