If you delete some active row, i.e. the row, whose auxiliary variable
is non-basic, the number of rows is decreased by one while the number
of variables marked as basic remains the same, and the basis becomes
invalid. Similarly, if you delete some basic structural variable, the
number of rows remains the same while the number of basic variables
is decreased by one, so the basis again becomes invalid.
If a row is active, deleting it invalidates the basis as was explained
above. However, you can attain the same effect by *freeing* the row,
i.e. by changing its bounds to -inf and +inf, resp., in which case the
basis remains valid. Analogously, if a column is basic, you can change
both its bounds to 0, i.e. *fix* the structural variable at zero, rather
than delete it, in which case the basis again remains valid.