I'm implementing a simplex method solver for a standard problem $$ \begin{aligned} \operatorname{minimize} \qquad&c^T x\\ \operatorname{subjected to} \qquad&Ax = b\\ &x \geq 0\\ \end{aligned} $$ To find basic feasible solution I'm adding new variables $y$ and solving additional problem $$ \begin{aligned} \operatorname{minimize} \qquad&\sum y\\ \operatorname{subjected to} \qquad&Ax + y = b\\ &x \geq 0\\ \end{aligned} $$ The last $m$ rows of matrix $(A\;E)$ form the basis. When the additional problem is solved the matrix $(A\;E)$ along with right hand side $b$ and set of basis rows are changed.
May the set of basis vaiables in the updated simplex tableau contain additional variables $y$? I've seen an example of such case if $A$ has incomplete rank, so assumming that $\operatorname{rank} A = m$.