# Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.

Given a finite dimensional vector space V, define LT(V):=span$_\mathbb{C}\{LT(w)|w\in V\}$.

I want to find a basis $\{v_1,\ldots , v_n\}$ for V such that LT(V)=span$_\mathbb{C}\{LT(v_1),LT(v_2),\ldots ,LT(v_n)\}$.

I have never worked with Gröbner bases and know very little about them, but this question looks similar to them I think, only here I'm talking about subspaces instead of ideals.

Is there a well-known method for finding such a basis? I've been searching, but haven't found anything yet, so I thought I'd ask on here.

For example, by an algorithm I know $\{1,t_1+t_3,t_2,t_1t_2,t_1t_2(t_1+t_3),t_2(t_1+t_3),t_2(t_2t_3),t_1t_2(t_2t_3)\}$ forms a basis for the space of sections $H^0(X,L)$ of of particular variety $X$ and a line bundle $L$ over it. And for the vector spaces I'm working with I already know $dim(LT(H^0(X,L))=dim(H^0(X,L))$, so I know $\{1,t_3,t_2,t_1t_2,t_1t_2t_3,t_2t_3,t_2(t_2t_3),t_1t_2(t_2t_3)\}$ forms a basis for $LT(H^0(X,L)$.

But in general, if the vector space I begin with is n-dimensional, it may not be the case that the basis I choose has n distinct lowest terms, so I'm looking for a way to find a basis with n distinct lowest terms.

• V is a subspace of the vector space on which your coordinate system is defined? Then a basis $\left(v_1, v_2, \ldots, v_n\right)$ has the property that you desire if and only if it is a permutation of the list of rows of a matrix in row-echelon form. I am fairly sure that this is well-known; many people motivate Groebner basis theory as a noncommutative analogue of classical linear algebra (Groebner basis <~~> row echelon form; reduced Groebner basis <~~> reduced row echelon form). Oct 7 '14 at 21:33