# Introduction to Linear Optimization: Driving the artificial variables out of the basis (case: no entries in the $j$-row are nonzero)

Reading the book Introduction to Linear Optimization by Bertsimas and Tsiklisis, I've come across the following subject: Driving the artificial variables out of the basis.

The description is as follows: suppose an artificial variable $x_{B(j)}$ is in the basis, then examining the $j$-row of the simplex tableau there are two cases:

Either the $j$-row of $B^{-1}A$ contains a nonzero element (I understand this case !), otherwise all elements in the row are equal to $0$, which imply that the rows of $A$ are linearly independent and it is shown that the constraint $a_j x = b_j$ is redundant.

The book then suggest we delete the $j$-row of the simplex tableau and continue from there. Why is this possible ? By deleting the $j$-row of the tableau, we remove the $j$-basic variable from the basis. Also, deleting the $j$-row of the simplex tableau corresponds to deleting the $j$-row of $A$, form the basis of the basic variables, not including the $j$'th basic variable (now one dimension smaller), and then forming the simplex tableau corresponding to these ?

Now, how can we be sure the vectors corresponding to the basic variables excluding the $j$'th are linearly independent and form a basis ($j$-element is removed from these vectors) ? How do I know the simplex tableau with the $j$-row removed is actually a real simplex tableau and what vectors and basis matrix does it correspond to ?

Please give me advice and tell me whether I'm wrong and what to do, I've been really thinking over this, but haven't come to a conclusion.