Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and write $\mathrm{span}(G_M)$ for the subspace of $k[x_0,\dots,x_{n-1}]$ formed by $k$-linear combinations of elements of $G_M$.
In general, the Gröbner basis $G_M$ varies depending on our choice of $M$. But in the small handful of examples I've checked, $\mathrm{span}(G_M)$ is the same subspace of $I$ for any choice of $M$, even when $G_M$ changes. Therefore I'm wondering if $\mathrm{span}(G_M)$ is always indepentent of our choice of $M$. Is this so? Or is there an example of $k$, $n$, and $I$ such that $\mathrm{span}(G_M)$ depends on the choice of $M$?
In general the number of elements in two Gröbner bases can be different and, so, they can span different vector spaces. For an easy example consider the ideal $(x^2+xy, y^2)$ of the ring $k[x,y]$ with lexicographic order. If $x>y$, then $\{x^2+xy,y^2\}$ is a reduced Gröbner basis, while, if $y>x$, the reduced Gröbner basis is $\{x^2+xy,y^2,x^3\}$.