Data Science from Scratch

Chapter 4. Linear Algebra

Is there anything more useless or less useful than Algebra?

Billy Connolly

Linear algebra is the branch of mathematics that deals with vector spaces. Although I can’t hope to teach you linear algebra in a brief chapter, it underpins a large number of data science concepts and techniques, which means I owe it to you to at least try. What we learn in this chapter we’ll use heavily throughout the rest of the book.

Vectors

Abstractly, vectors are objects that can be added together (to form new vectors) and that can be multiplied by scalars (i.e., numbers), also to form new vectors.

Concretely (for us), vectors are points in some finite-dimensional space. Although you might not think of your data as vectors, they are a good way to represent numeric data.

For example, if you have the heights, weights, and ages of a large number of people, you can treat your data as three-dimensional vectors (height, weight, age). If you’re teaching a class with four exams, you can treat student grades as four-dimensional vectors (exam1, exam2, exam3, exam4).

The simplest from-scratch approach is to represent vectors as lists of numbers. A list of three numbers corresponds to a vector in three-dimensional space, and vice versa:

height_weight_age = [70,  # inches,
                     170, # pounds,
                     40 ] # years

grades = [95,   # exam1
          80,   # exam2
          75,   # exam3
          62 ]  # exam4

One problem with this approach is that we will want to perform arithmetic on vectors. Because Python lists aren’t vectors (and ...