Practical Linear Algebra for Data Science

Chapter 3. Vectors, Part 2

The previous chapter laid the groundwork for understanding vectors and basic operations acting on vectors. Now you will expand the horizons of your linear algebra knowledge by learning about a set of interrelated concepts including linear independence, subspaces, and bases. Each of these topics is crucially important for understanding operations on matrices.

Some of the topics here may seem abstract and disconnected from applications, but there is a very short path between them, e.g., vector subspaces and fitting statistical models to data. The applications in data science come later, so please keep focusing on the fundamentals so that the advanced topics are easier to understand.

Vector Sets

We can start the chapter with something easy: a collection of vectors is called a set. You can imagine putting a bunch of vectors into a bag to carry around. Vector sets are indicated using capital italics letters, like S or V. Mathematically, we can describe sets as the following:

$$V=\{{𝐯}_{\mathbf{1}},...,{𝐯}_{𝐧}\}$$

Imagine, for example, a dataset of the number of Covid-19 positive cases, hospitalizations, and deaths from one hundred countries; you could store the data from each country in a three-element vector, and create a vector set containing one hundred vectors.

Vector sets can contain a finite or an infinite number of vectors. Vector sets with an infinite number of vectors may sound like a uselessly silly abstraction, but vector subspaces are infinite vector ...