The idea of seeing a "geometry" as the study of invariants under certain transformation groups is certainly an interesting one (so for example Euclidean geometry is the field of math that studies figures that are indistinguishable under rigid motions, whereas in topology we also allow the broader equivalence relation of homeomophism)... but is this all?
Linear algebra and vector spaces are often seen as part of geometry, too; and differential geometry looks just like calculus but the name itself points at geometry instead. Not to mention matrices and determinants, and sometimes even the methods for solving a linear system, which should be purely algebraic concepts.
I know tradition probably plays a big role in why certain topics are seen as "alegbra" while others are seen as "geometry" or "calculus": but still, is there a widely accepted definition of geometry that explains why the branches I mentioned are considered to belong to geometry?