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I learned the objective variable can be written as: $$ z = c_B^T \bar{b} - \xi^Tx $$ where $\bar b = B^{-1} b$ and $\xi$ is the "indicator vector" which is usually put at the top or bottom of simplex tableau.

But I do not understand why each iteration, the row operation to unset (zero) previous $\xi_B$ will produce the correct new $\xi$.

In my textbook, the $\xi$ has zeros in its first $m$ element and $\xi_j = c_B^T \bar{A}_j - c_j$ when $j = m+1, m+2 ... n$.

I understand initially if $c_B^T = \mathbf{0}$ then $\xi_j = -c_j$ and the row operation to zero previous $\xi_B$ is essentially add $c_B^T \bar A_j$ onto it. So in the first iteration I can successfully reason why row operation produces $\xi_j = c_B^T \bar{A}_j - c_j$. But after the first iteration, $c_B^T \not= \mathbf{0}$ then why the row operation performed on objective row will produce the $\xi_j = c_B^T \bar{A}_j - c_j$?

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