Definition of Trace of Linear Operator

Another common definition is the sum of eigenvalues (in the algebraic closure), counted with multiplicity. Since the eigenvalues are algebraic (they satisfy $A$'s minimal polynomial) we can invoke Galois theory to show the sum, being invariant, is in the original scalar field. By tensoring the space against the algebraic closure of the scalar field ("extension of scalars") and then writing it as a sum of generalized eigenspaces, corresponding to Jordan blocks of $A$, we can show that the sum of eigenvalues is equal to the sum of diagonal entries of $A$ in some basis (granted, after we extend the scalars), which we can then show is invariant under basis-change.

Another way is if we write ${\rm tr}:{\rm End}(V)\cong V\otimes_F V^*\to F$, where $v\otimes f\mapsto f(v)$ in the obvious way, and the expression $V^*$ denotes the dual vector space (i.e. $\hom_F(V,F)$). By choosing an ordered basis we can show this is the same as summing the diagonal entries in that basis. This fits into the perspective of string diagrams, a nice visual language for describing tensor facts, including currying and ${\rm tr}(AB)={\rm tr}(BA)$. Proving identities means wiggling strings around, yay!.