My question is about improving a basic feasible solution according to Bazaraa's explanation (Linear Programming and Network Flows ed II p94)
Improving a basic feasible solution
$z_{0}=c_{B}B^{-1}b$
$x_{B}=B^{-1}b-B^{-1}Nx_{N}=B^{-1}b-\sum_{j \epsilon R}^{} B^{-1}a_{j}x_{j}$
where R is the current set of the indices of the nonbasic variales
$z = cx = c_{B}x_{B} + c_{N}x_{N} = c_{B}(B^{-1}b - \sum_{j \epsilon R}^{} B^{-1}a_{j}x_{j}) + \sum _{j \epsilon R}^{}c_{i}x_{j} = z_{0}-\sum_{j \epsilon R}^{}(z_{j}-c_{j})x_{j}$
where $z_{j}=c_{B}B^{-1}a_{j}$ for each nonbasic variable.
In conclusion:
$z=z_{0}-\sum_{j \epsilon R}^{}(z_{j}-c_{j})x_{j}$
Ideas
I really can't understand what's the idea behind $z_{j}$. First of all the formula for $z_{j}$ doesn't look like the one for $z_{0}$. Second, why in the formula we have a mix corresponding to basic variables ($c_{B}$, $B^{-1}$) and nonbasic variables ($a_{j}$)