# Understanding the steps of Karmarkar's algorithm

I am working through Karmarkar's seminal paper [0] and came across something I didn't quite understand. In section 2.3, Description of the Algorithm, he explains how to calculate the next point. The point $$\mathbf{x^{(k)}} \in \mathbb{R}^n$$ that is put into $$\phi$$ has to be a feasible point, i.e., A$$\mathbf{x}^{(k)}=0$$. The matrix $$D$$ is a diagonal matrix with the $$\mathbf{x_i}^{(k)}$$ on the diagonal. Now,

\begin{align*}AD=\begin{pmatrix}A_1 \mathbf{x}_1^{(k)}\dots A_n \mathbf{x}_n^{(k)} \end{pmatrix}\end{align*}

where $$A_i$$ is the $$i$$th column of $$A$$. We know that the sum of these components is $$0$$.

The matrix $$B$$ is the matrix $$AD$$ with a row of $$1$$'s at the bottom. Now, in the algorithm it says that

$$\mbox{Ker} B \subseteq \Sigma = \left\{ \mathbf{x} \mid \sum_i x_i=1 \right\}$$

But $$\mbox{Ker} B = \vec{0}$$, because the last row of $$B$$ consists of ones. Am I not seeing something here?

[0] Narendra Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, Volume 4, Issue 4, December 1984.