Cramer's rule
2019-03-18
Grant Sanderson uploaded a new video the other day explaining Cramer’s rule by drawing on the connection between determinants and parallelepiped areas. I enjoyed the explanation, and it motivated me to express the derivation in the language of differential forms, which would make the parametrization independence more obvious for statements about area that on first glance appear to have some parametrization dependence.
Like Grant did, I’ll write everything out for a two-dimensional problem, and you can test your understanding by generalizing it to arbitrary dimensions. 😉
The equation we want to solve is The forms and are vectors in a two dimensional vector space with basis , and is a linear map on that vector space. Because of the antisymmetry of the wedge product, we can get the components of by taking wedge products with the basis vectors. Choosing to write in component form as we see that and
The form is a vector in a one-dimensional vector space. There’s a useful linear map to define on this vector space which simply applies a linear operator to each of the vectors in the wedge product: Since the only linear map on a one-dimensional vector space is multiplication by a scalar, this map multiplies by a scalar that depends on . That scalar is the determinant of , which from this definition is clearly independent of any parametrization of the forms used:
Now let’s play around with our equation a bit to try and get determinants into the picture. Start by wedging on the right of both sides of the equation: The left-hand side looks promising, so develop it further: Now what to do about the right hand side? Let’s define two new linear maps by their action on the basis vectors and : and These are the equivalent of replacing different columns in that you find in Grant’s derivation and the presentation on Wikipedia. The nice thing about these maps is that they let us get a determinant on the right-hand side of our wedged equation: Now we’ve solved for the component! The component is obtained in an analogous way using instead of .
One aspect I like about using forms is that the arbitrariness of what measure one uses to compute areas clearly cancels out. Different measures will assign different areas to , but it doesn’t matter since the same factor appears on both sides of the equation.