# Normals and the Inverse Transpose, Part 1: Grassmann Algebra

A mysterious fact about linear transformations is that some of them, namely nonuniform scalings and
shears, make a puzzling distinction between “plain” vectors and normal vectors. When we transform
“plain” vectors with a matrix, we’re required to transform the normals with—for some
reason—the *inverse transpose* of that matrix. How are we to understand this?

It takes only a bit of algebra to show that using the inverse transpose ensures that transformed normals will remain perpendicular to the tangent planes they define. That’s fine as far as it goes, but it misses a deeper and more interesting story about the geometry behind this—which I’ll explore over the next few articles.