  We prove that on $\mathbb{R}^n$, there is no $N$-supercyclic operator with
$1\leq N< \lfloor \frac{n+1}{2}\rfloor$ i.e. if $\mathbb{R}^n$ has an $N$
dimensional subspace whose orbit under $T$ is dense in $\mathbb{R}^n$, then $N$
is greater than $\lfloor\frac{n+1}{2}\rfloor$. Moreover, this value is optimal.
We then consider the case of strongly $N$-supercyclic operators. An operator
$T$ is strongly $N$-supercyclic if $\mathbb{R}^n$ has an $N$-dimensional
subspace whose orbit under $T$ is dense in $\mathbb{P}_N(\mathbb{R}^n)$, the
$N$-th Grassmannian. We prove that strong $N$-supercyclicity does not occur
non-trivially in finite dimension.
