I have two 6x1 element basis vectors $s_1$ and $s_2$ defined in local coordinates which can be combined into a single 6x2 subspace of $ s = \begin{vmatrix} X_{2}s_{1} & s_{2}\end{vmatrix}$ where $X_2$ is a 6x6 affine transformation matrix.
If I know the individual 6x5 nullspaces $R_1 = {\rm null}(s_1^\top)$ and $R_2 = {\rm null}(s_2^\top)$ such that $s_1^\top R_1 = 0$ and $s_2^\top R_2 = 0$ how do I calculate the combined 6x4 nullspace $R={\rm null}(s^\top)$ from $R_1$ and $R_2$ ?
Example
A joint is described by the rotation $s_1^\top = \begin{vmatrix} 0&0&0&0&0&1 \end{vmatrix}$ followed by the translation $s_2^\top = \begin{vmatrix} 1&0&0&0&0&0 \end{vmatrix}$ with the affine transformation $ X_{2}=\begin{vmatrix}1 & & & 0 & -z & y\\ & 1 & & z & 0 & -x\\ & & 1 & -y & x & 0\\ & & & 1\\ & & & & 1\\ & & & & & 1 \end{vmatrix} $
Combined this gives $ s = \begin{vmatrix}-y & 1\\ x & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 1 & 0 \end{vmatrix} $
The computed nullspace with Matlab(r) is $ R = {\rm null}(s^\top)=\begin{vmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{1}{x}\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{vmatrix} $
The question is how do I arrive at this algorithmically or analytically?