Abstract -- We give a formula for determining when a set of k vectors in n-space is linearly independent, and if so, what is the volume of the parallelepiped with these vectors as its sides. This function, the gramian, allows one to partially apply the determinant when the number of vectors you have is less than the dimension of the ambient space. These results were classically known, but are not part of the standard linear algebra curriculum. Prerequisites: familiarity with matrices and determinants.