# Basic

### From Glossary

Associated with a submatrix of , say , whose columns comprise a
basis for (i.e., consists of linearly independent columns
of , which is a *basis* for ).

Here are some related terms that arise in linear programming.

*Adjacent basis*. One that differs in exactly one column from a given basis.*Basic column*. A column of the basis matrix.*Basic variable*. The variable, say , associated with the -th column of the basis matrix.*Basic level*. The value of a basic variable.-
*Basic solution*. The solution, , obtained by setting nonbasic values to some bound value, like 0, resulting in a unique solution for the basic variables. That is, is equivalent to , where and . Upon fixing the value of , the nonsingularity of gives the basic solution with . In a standard linear program, , and hence . *Basic feasible solution*. A basic solution that is feasible --i.e., the basic values satisfy their bounds. (In a standard LP, this means .)*Basis kernel*. After performing forward triangularization, if the basis does not triangularize completely, backward triangularization is applied. The result is a (rearranged) blocking of the basis into three segments:|\ | \ <--- Forward triangle |__\ ______ | | | | | | <--- Kernel | |______| | |\ | | \ <--- Backward triangle |__________|__\

Each row and column in the kernel has at least 2 nonzeroes.