Let $V$ be a vector space and $dim(V)=n$.
Let $GL(V)$ denote the set of all invertible linear transformations from $V$ to itself.
Next we define $Gr(k,n)$ to be set of all $k-$dimensional subspaces of $V$ with $k<n$.
We now define the following action:
$GL(V) \times Gr(k,n) \to Gr(k,n)$ defined as $T.W\mapsto T(W) $
I'm asked to show that the action is transitive.
My thoughts: To show that the action is transitive we must show that, given $U$ $\in Gr(k,n)$ $\exists\;\; T'\in GL(V)$ such that $T'.W=U$
I know that if $W$ is a subspace of $V$ and if $T\in GL(V)$ then $T(W)$ is a subspace of $V$ and $dim(T(W))=k$
So basically as $T$ varies in $GL(V)$ so does the $T(W)$ vary in $V$. But how can I show that every $k-$dimensional subspace of $V$ is the image of some transformation?