# Finding an orthogonal basis outside the intersection of two subspaces.

Let $A_1 \in \mathbb{C}^{10 \times 200}$, $A_2 \in \mathbb{C}^{10\times 200}$ and let $G_1 \in \mathbb{C}^{200\times 190}$ represent an orthogonal basis for $N(A_1)$, $G_2 \in \mathbb{C}^{200\times 190}$ be an orthogonal basis for $N(A_2)$. I computed the intersection of the two null spaces and found out that they overlap in $180$ dimensions (i.e, $\dim(N(A_1)\cap N(A_2))=180$). Let $G_3 \in \mathbb{C}^{200\times 180}$ be an orthogonal basis for the subspace $N(A_1)\cap N(A_2)$, here is my question:

How can I find two orthogonal basis $Z_1 \in \mathbb{C}^{200 \times 10}$ and $Z_2 \in \mathbb{C}^{200 \times 10}$ such that $Z_1$ is part of the basis $G_1$ but not in $N(A_1)\cap N(A_2)$ and $Z_2$ is part of the basis $G_2$ but not in $N(A_1)\cap N(A_2)$. Additionally I want $Z_1$ and $Z_2$ to be orthogonal, i.e $Z_1^HZ_2=0$.

What I've done so far: to find $Z_1$, I computed the row space of $G_3$ ($G_3$ is the basis that spans the subspace $N(A_1)\cap N(A_2)$), this row space is outside $N(A_1)\cap N(A_2)$), and then computed the intersection of this row space with $G_1$ ($G_1$ is the basis for $N(A_1)$). I did the same to find $Z_2$, but here I computed the intersection of the row space with $G_2$, ($G_2$ is the basis for $N(A_2)$). My only problem is that I'm NOT getting $Z_1^HZ_2=0$, but $Z_1^HZ_2$ is small nonetheless (values in the matrix resulting from $Z_1^HZ_2$ are around $10^{-3}$). Any help is much appreciated.