When talking about vector space, you usually need the “scalar (field)”, and scalars need inverse to be well-defined.
So for integers, the scalar should be integer itself.
Sadly, inverse of integers stops being an integer, from where all sorts of number theoretic nightmare occurs
Instead, integers form a ring, and is a module over scalar of integers.
A vector space is when you can:
And get another Thing that’s the same Kind of Thing.
By Thing I mean Vector and by Kind of Thing I mean element of the same Vector Space.
Examples of vector spaces:
Examples of Not Vector Spaces:
Yeah a few of these come with asterisks I’m happy to answer questions but don’t want to argue with pedants.
wow didn’t expect this to be so general. How do integers not fit into the definition ? you can add them together and obtain another integer
When talking about vector space, you usually need the “scalar (field)”, and scalars need inverse to be well-defined.
So for integers, the scalar should be integer itself. Sadly, inverse of integers stops being an integer,
from where all sorts of number theoretic nightmare occursInstead, integers form a ring, and is a module over scalar of integers.