I am learning the Simplex Algorithm, and currently understand every step except one part. The relevant section from my notes are below. In the second step below, I am not sure why we need to pick $a_{ij}$ to minimise $a_{i0}/a_{ij}$.
My understanding is that, the coefficients of the bottom row should be $0$ for variables in the basis. Therefore, once we have found the variable with positive coefficient in the bottom row to enter our basis, and found any constraint row where its coefficient is non-zero, then we will improve our solution by pivoting on such a row, are increasing a variable with positive coefficient in the objective function, and decreasing a variable with coefficient $0$ in the basis function. Is this step then just for efficiency?
Thank you very much for your time :)
