In class I learned the following pseudocode for the simplex algorithm:
However, what is unclear to me about this procedute is the "solve" command in lines 3 and 8. When solving $A_B^Ty = c_B$, do the matrix $A_B^T, c_B$ need to be changed too, i.e. do the row transformations done in the Gauss algorithm used to solve $A_B^Ty = c_B$ carry over to the whole algorithm? Could you please explain this to me?
For a given $B$, in the simplex method we have that $A_B$ is invertible. So in this case we are solving for $y$ via $y = A_B^{-1}c_B$. Then we update $B$ later in the loop.