chapter six

6 Generalizing to Higher Dimensions

This chapter covers

Implementing a Python abstract base class representing general vectors
Defining vector spaces and listing their useful properties
Interpreting functions, matrices, images, and sound waves as vectors
Finding useful subspaces of vector spaces, containing data of interest

Even if you’re not interested in animating teapots, the machinery of vectors, linear transformations, and matrices can be useful. In fact, these concepts are so useful there’s an entire branch of math devoted to them: linear algebra. Linear algebra generalizes everything we know about 2D and 3D geometry to study different kinds of data having any number of dimensions.

As a programmer, you’ve already got experience with generalization. When writing a piece of software, it’s common to find yourself writing similar code over and over. At some point you catch yourself doing this, and you consolidate the code into one class or function capable of handling all of the cases you’ve seen. This saves you typing and often improves code organization. Mathematicians follow the same process: after encountering similar patterns over and over, they can better state exactly what their seeing and refine their definitions.

6.1 Generalizing our definition of vectors

6.1.1 Creating a class for 2D coordinate vectors

6.1.2 Improving the Vec2 class

6.1.3 Repeating the process with 3D vectors

6.1.4 Building a Vector base class

6.1.5 Defining vector spaces

6.1.6 Unit testing vector space classes

6.1.7 Exercises

6.2 Exploring different vector spaces

6.2.1 Enumerating all coordinate vector spaces

6.2.2 Identifying vector spaces in the wild

6.2.3 Treating functions as vectors

6.2.4 Treating matrices as vectors

6.2.5 Manipulating images with vector operations

6.2.6 Exercises

6.3 Looking for smaller vector spaces

6.3.1 Identifying subspaces

6.3.2 Starting with a single vector

6.3.3 Spanning a bigger space

6.3.4 Defining the word “dimension”