General linear group over subspace is conjugacy-closed

Statement
Suppose $$V$$ is a finite-dimensional vector space over a field $$k$$, written as a direct sum of subspaces $$U$$ and $$W$$. Then, consider the map:

$$i: GL(U) \to GL(V)$$

that sends a linear map on $$U$$ to a linear map on $$V$$ that behaves the same way on $$U$$ and is the identity on $$W$$. This is an injective homomorphism, hence we can identify $$GL(U)$$ with a subgroup of $$GL(V)$$.

Then, if $$A,B \in GL(U)$$ are such that the images $$i(A)$$ and $$i(B)$$ are conjugate in $$GL(V)$$, then $$A,B$$ are conjugate in $$GL(U)$$.

Other related facts

 * Brauer's permutation lemma
 * General linear group over subfield is conjugacy-closed
 * Symmetric group on finite or cofinite subset is conjugacy-closed