# Determinant Intuition in Abstract Vector Spaces over Arbitrary Fields

I was wondering if there was an intuitive way to think about determinants of matrices that represent linear transformations in abstract vector spaces over arbitrary fields.

There are many posts about the intuition for the determinent of n-tuple spaces over the field of real numbers where they represent how the volume of the shape outlined by the basis vectors change and about how the determinant is negative should two basis vectors "flip about". There is lots on this here: What's an intuitive way to think about the determinant?

However I am wondering how the idea of a determinant generalities to more abstract vector spaces over arbitrary fields. Does it actually mean something intuitively or does it just become some computation. I suppose that a larger determinant simply means that the basis vectors are transformed to carry more "magnitude" of some sort. More importantly what would it mean to have a negative determinent in this context, i.e. for the basis vectors of a more abstract space such as that of polynomials to "flip about" as they would for classical n-tuple vectors.

What is the best mindset to approach this topic as I am learning it?

• I've also thought of determinant as some "linear dependence measure": in many cases the most important information it carries is whether it is zero or not. As a universal alternating linear map, its value, if non-zero, does not say much if we allow rescaling; but vanishing of determinant always implies linear dependence, even in arbitrary field. Dec 14 '17 at 0:44

I don't know if this is the answer you are looking for, but the term 'intuition' is very obscure and undefined, so I'll try anyway. I like thinking of determinant as an invariant of matrices. By this I mean that the determinant gives us sort of classification scheme of matrices. You probably know that two matrices $A,B$ are called similar if there exist a non-singular matrix $P$ such that
$$A=PBP^{-1}$$
In this case, you get as a consequence that also $\det A=\det B$. In other words, $\det A$ is invariant under similarity transformations $A\mapsto PAP^{-1}$.
This is very much like the genus of surfaces: a topological classification scheme that classifies surfaces by counting 'holes'. This kind of thinking might be too abstract, but I am not sure you can in general get geometric intuition like 'a measure of volume' as you can get in $\mathbb{R}^{n}$. Nevertheless, I think this kind of point of view is very important. It shows up over and over again everywhere in mathematics and physics.