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.



Is there a Gröbner basis analogue that exists for vector spaces?
Darij is right. Computing Groebner basis for a linear ideal is the same as computing the row echelon form (this can be seen from the Buchberger algorithm). And I think he meant "nonlinear" instead of "noncommutative". 


