Over $\mathbb R^n$, the standard intuition given to the determinant is that it measures the signed area of the image of an unit cube. But determinants can be more generally defined for endomorphisms of spaces of finite dimension over other fields than $\mathbb R$, where this notion of area doesn't exist, or even over other rings, or semirings or other things. Is there more general intuition for what the determinant means that isn't specific to the field $\mathbb R$?
Asked
Active
Viewed 60 times
2
-
1You could start by reading about the exterior algebra on a vector space and how determinant is the scalar $$ \bigwedge^n f: \bigwedge^n V \to \bigwedge^n V, $$ $v_1 \wedge \cdots \wedge v_n \mapsto (\det f), v_1 \wedge \cdots \wedge v_n$ associated to a linear map $f: V \to V$ when $\dim V = n$. – Sammy Black Aug 07 '24 at 19:25
-
My apologies for the closure; I looked for duplicates and essentially this question was asked several years ago at https://math.stackexchange.com/questions/2565713/determinant-intuition-in-abstract-vector-spaces-over-arbitrary-fields?rq=1, although I don't think the accepted answer there really answers the question, so I'll add another answer over there. – Qiaochu Yuan Aug 07 '24 at 22:38