I'm new to topology, I'm struggling with how the dimension of a (real) Grassmannian Gr(k,n) is defined?
I know the definition of dimension of a vector space, but Gr(k,n) does not seem like a vector space (if it is what is the vector addition and over which field is it a vector space?)
In such a case, how is the dimension of Gr(k,n) defined (could you provide a definition)?