Show that the variable that becomes nonbasic in one iteration of the simplex
method cannot become basic in the next iteration.
I think I'm confused because we have been using dictionary notation, not matrix notation. I understand that the column that corresponded to the variable that just left is now 0 because that variable is now 0. Can we say that the variable will never be basic again because it will have the wrong sign on the coefficient in the objective function after it has exited the basis?
I see. Time to back up to the geometry. What are the requirements or assumptions for a solution to exist?
Any of these things lead in the right direction?
In other words, as you just created a nonbasic variable, because it was a maximal constraint, the choice to return it to basic on the next iteration would seem to indicate that we didn't use it all.