Can sage extend bases?
Say I have subspaces $X\subset Y$ of a vector space $V$. I want to extend a basis for $X$ to a basis for $Y$. Does sage have this capability?
For example, I have two matrices
sage: d1 = matrix(GF(2), [ ....: [1,0,1,0,1,0,1,0,1,0], ....: [1,1,0,0,0,0,0,0,0,1], ....: [0,1,1,1,0,0,0,0,0,0], ....: [0,0,0,1,1,1,0,0,0,0], ....: [0,0,0,0,0,1,1,1,0,0], ....: [0,0,0,0,0,0,0,1,1,1] ....: ]) sage: d2 = matrix(GF(2), [[1,1,1,0,0,0,0,0,0,0],[0,0,1,1,1,0,0,0,0,0]]).transpose()
Since d1*d2=0 we know that the column space of d2 is a subspace of the null space of d1. We can obtain a basis for the column space of d2 with
How could we extend this basis to a basis of the null space of d1?