The prior section develops the determinant algebraically, by considering what formulas satisfy certain properties. This section complements that with a geometric approach. One advantage of this approach is that, while we have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. (The prior section handles determinants as functions of the rows, but in this section columns are more convenient. The final result of the prior section says that we can make the switch.)