 Basic feasible solution

For a full-row-rank system $Ax=b$ with $m$ rows and variables $x\ge 0$, choose $m$ linearly independent columns as a simplex basis. Set all nonbasic variables to zero and solve $x_B=A_B^{-1}b$. If $x_B\ge 0$, the resulting vector is a basic feasible solution. Redundant equality rows can first be removed.

Equivalently, a feasible vector is basic when the columns indexed by its strictly positive coordinates are linearly independent. Those columns can be extended to a basis of the full column space because $A$ has row rank $m$. The extension may have zero-valued basic variables; this is degeneracy in linear programming.

For $x_1+x_2=1$ and $x_1,x_2\ge 0$, the two basic feasible solutions are $(1,0)$ and $(0,1)$. The feasible midpoint $(1/2,1/2)$ is not basic: its two positive columns are dependent. In this standard form, basic feasible solutions are exactly the extreme points of the feasible polyhedron, connecting basis algebra to the Fundamental theorem of linear programming.