Where I've written the vector with $x_B, x_N$ here that should be considered as a partition between them (there should be a horizontal dashed lines or something between them).
\begin{align*} f(x_{n \times 1}) &= c^T x_{n \times 1} \\ &= [ c_B^T \vert c_D^T] \begin{bmatrix} x_B \\ x_N \end{bmatrix} \end{align*}
Subject to
\begin{align*} A_{m \times n} x_{n \times 1} = b_{m \times 1} \end{align*}
Partition $A$ as $A = [B \vert D]$ where $B$ is a square $m \times m$ matrix and $D$ is $(n-m) \times m$.
If $x_N$ are non-basic then they're all zero can we can read off the BFS as
\begin{align*} B x_b &= b \\ x_b &= B^{-1} b \\ \end{align*}
For BFS
\begin{align*} x = \begin{bmatrix} x_B \\ 0 \end{bmatrix} = \begin{bmatrix} B^{-1}b \\ 0 \end{bmatrix} \geq 0 \end{align*}
For the cost function the formula is
\begin{align*} f(x) &= c_B^T x_B + c_D x_N \\ \end{align*}
Then
\begin{align*} c_B^T B^{-1} b = z_0 \end{align*}
\subsection{What happens if some $x_N$ increases from zero}
If some element of $x_N$ increases from zero then the non-zero component of $x_B$ must drop by $B^{-1}dx_N$ in order to maintain the equality constraint.
\newpage \subsection{Question}
I'm struggling to understand the simplex method and what happens when a variable from the non-basic set is brought into the basic set, and from basic to non-basic.
For the simplex method the cost function can be written as
\begin{align*} f(x) &= c_B^T x_B + c_D^T x_N \\ \end{align*}
Where $B$ relates to the variables in the basic solution and D to those in the non-basic.
This cost function is subject to
\begin{align*} A_{m \times n} x_{m \times 1} + = b_{m \times 1} \end{align*}
$A$ can be rewritten as
\begin{align*} A = [B \vert D] \end{align*}
Where $B \in \mathbb{R}^{m \times m}$ and $D \in \mathbb{R}^{(n -m) \times m}$.
We can write this as
\begin{align*} \begin{bmatrix} B \vert D \end{bmatrix} \begin{bmatrix} x_B \\ x_N \end{bmatrix} = Bx_B + Dx_N \end{align*}
For the basic feasible solution $x_N = 0$, so we have
\begin{align*} Bx_B + Dx_N = Bx_B = b \end{align*}
When we go through the simplex we take some zero element from $x_N$ and increase its value, but this means that something has to happen to the corresponding row in $x_B$ to maintain the equality.
I have it written down that $x_B$ must "drop in value by $B^{-1}dx_N$ to maintain equality".
I don't understand how this works out though
What I want to know is how much I have to bring down the values in $x_B$ when a value from $x_N$ is increased, why that value is true, and a concrete example for it.