Part six of the Linear Algebra for Quantum Computing series is here and it finally gets us through the concept of vector spaces! Or at least, gets us to a place where we can understand the formal “mathematician’s” definition of a vector space. This sheet makes references to sheet five, which if you haven’t checked out yet can be found here.
In sheet six, we cover what a field is and what it means for something to be an action of a field. Together with the concept of an abelian group introduced in sheet five, we have all the components we need to understand what a vector space is.
Below, I’m also including a simple graphic I created that shows which “tests” need to be “run” (sorry, I think of things in software terms) in order to determine whether something is a vector space over a field. I’ve categorized them by where the input values for the tests come from - the abelian group or the field or both. This helped me when I was having trouble remembering which tests required what. Hope it helps you too!
UPDATE 08.31.2022: Since these sheets are now complete, styled, reviewed, edited, and published, you can pick up the whole pack of 15 in my shop here!