We consider the constraint equation is $Ax=b$ where $A$ is a $m \times n$ matrix.
I know that in phase one of the Simplex method, we should find a vertex/corner of feasible set.
I'm confused that why the corner must have $m$ nonzero components and $n-m$ zero components?
How to understand VERTEX/CORNER from the perspective of geometry and algebra?
Each equation in the constraint is one hyperplane that represents one face of the constraint set polytope.
When you are on one face, the corresponding equation is solved to equality (because all points on the hyperplane solve the equation of the hyperplane itself).
If you are on a corner of two faces, you can think of each face individually as a hyperplane, then the point is on both hyperplanes, and you see that two equations must be solved to equality.
You can generalize this to $m$ hyperplanes and you see that all the equations of all the hyperplanes that create the corner or vertex your point is in have to be solved to equality, so they have to be zero.