Chapter 3: Foundations

Up until this point, we have introduced our mathematics with as little rigor as possible. This chapter – and this part of the book – will change that. You may ask why? Well, this rigor and foundational material is needed when we get to much more complex concepts such as Hilbert spaces and tensor products. Without this chapter, these advanced concepts will not make sense, and you won't have the context to understand them.

This chapter goes through the field of abstract algebra. As you might expect, there will be some abstract concepts that will be explored. Abstract algebra takes a step back from all other forms of algebra, such as linear, Boolean, and elementary algebra, and it tries to see what can be generalized between them. Mathematicians have found that they can generalize a few foundational concepts that, when put together, allow us to go further in math than we have before and help us understand it at a more fundamental level. Within...