7.1 Special transformations
So far, we have aspired to develop a geometric view of linear algebra. Vectors are mathematical objects defined by their direction and magnitude. In the spaces of vectors, the concept of distance and orthogonality gives rise to a geometric structure.
Linear transformations, the building blocks of machine learning, are just mappings that distort this structure: rotating, stretching, and skewing the geometry. However, there are types of transformations that preserve some of the structure. In practice, these provide valuable insights, and additionally, they are much easier to work with. In this section, we will take a look at the most important ones, those that we’ll encounter in machine learning.
7.1.1 The adjoint transformation
In machine learning, the most important stage is the Euclidean space ℝn. This is where data is represented and manipulated. There, the entire geometric structure is defined by the inner product
giving rise to the notion...
