  In this short note, we investigate some features of the space $\Inject{d}{m}$
of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss
in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this
context, as the set of orthonormal systems of $d$ vectors in $\bbR^m$. Finally,
we show that the Stiefel manifold $V_{m,d}$ is a deformation retract of
$\Inject{d}{m}$. One possible application of this remarkable fact lies in the
study of perturbative invariants of higher-dimensional (long) knots in
$\bbR^m$: in fact, the existence of the aforementioned deformation retraction
is the key tool for showing a vanishing lemma for configuration space integrals
{\`a} la Bott--Taubes (see \cite{BT} for the 3-dimensional results and
\cite{CR1}, \cite{C} for a first glimpse into higher-dimensional knot
invariants).
