8.4 Summary

With the study of the connections between linear algebra and graph theory, our journey through linear algebra is over.

In this and the previous seven chapters, we have learned that vectors and matrices are not merely data structures that store observations and measurements. Vectors and matrices possess a rich and beautiful geometric structure, describing data and their transformations at the same time!

First, we learned that vectors live in so-called vector spaces, the high-dimensional generalizations of the three-dimensional space we are living in (which might be 26-dimensional, according to some string theorists, but let’s stick to the Earth for now). We can measure lengths and distances via norms, most often defined by

or measure angles (among others) via inner products, most often defined by

From a mathematical perspective, matrices originate from the linear transformation of vector spaces, i.e., functions of the form f : U →V , satisfying the linearity...