What exactly is a basis matrix in LPP?

Question

I'm studying simplex for solving a LPP. $A$ is a $m\times n$ matix. $$Ax=b$$Then the book says:

Let $x$ be a basic feasible solution to the standard form problem. Let $$B(1), B(2),\cdots B(m)$$ be the indices of the basic variables and let $$B=[A_{B(1)}\cdots A_{B(m)}]$$ be the corresponding basis matrix.

I can't understand what's this $B$ a basis for. Thanks!

Answer 1

Note that the columns of $B$ are independent and the columns of $B$ indeed span $\mathbb{R}^m$.

We can always solve $$Bx_B=b$$ for any given $b$.