Basic Variables
https://people.math.carleton.ca/~kcheung/math/notes/MATH5801/04/4_2_bfs.html
They are called "basic" because they are the variables belonging to the Basis of the solution.
Assume a matrix \(A\) has rank \(m\). If \(Ax = b\) has a solution then there exists a solution that has at most \(m\) non zero variables. Because, there are \(m\) linearly independent columns in \(A\), and since any set of \(m\) linearly independent columns gives a basis in \(\mathbb{R}^m\), there exists a combination of those \(m\) columns that equals \(b\). The \(m\) columns form a basis and those variables are called basic variables.
- The basis need not be unique
- For every basic, the solution is unique
