# Normals and the Inverse Transpose, Part 2: Dual Spaces

In the first part of this series, we learned about
Grassmann algebra, and concluded that normal vectors in 3D can be interpreted as bivectors. To
transform bivectors, we need to use a different matrix (in general) than the one that transforms
ordinary vectors. Using a canonical basis for bivectors, we found that the matrix required is the *cofactor
matrix*, which is proportional to the inverse transpose. This provides at least a partial
explanation of why the inverse transpose is used to transform normal vectors.

However, we also left a few loose ends untied. We found out about the cofactor matrix, but we didn’t really see how that connects to the algebraic derivation that transforming a plane equation $N \cdot x + d = 0$ involves the inverse transpose. I just sort of handwaved the proportionality between the two.