Gröbner base for some lexicographic order

Let $\displaystyle A(a_{ij})$ a non null real matrix of dimension $\displaystyle m \times n$. Let $\displaystyle I$ the ideal generated by the $\displaystyle m$ polynomials $\displaystyle \sum_i a_{ji}x_i$. Show that this polynomials are a Gröbner base of the ideal I for some lexicographic order.

Thanks!!

Re: Gröbner base for some lexicographic order

Consider the lexicographic order $\displaystyle x_1<x_2<\ldots$.

Reduce the matrix in row echelon form using row operations; The polynomial set produces the same ideal, and cannot be reduced any further.

Thus, this set (and equivalently the starting set) is a Groebner basis.