Verifying a linear transformation

[Mr Davis 97]

I am told that the trace function tr(A) is a linear transformation. But this function maps from the space of matrices to the real numbers. How can this be a linear transformation if the set of real numbers isn't a vector space? Or is it? Can a field also be considered a vector space?

 

[Stephen Tashi]

Can a field also be considered a vector space?

Yes.

To split hairs , the "field of real numbers" is technically different than "the vector space of real numbers", but, Platonically, we think of the real numbers as being a single "thing" and that thing satisfies the properties of a vector space if we define the set of vectors as being the real numbers, the set of scalars as being the real numbers and the zero vector and the zero scalar, each to be the number zero.

 

[fresh_42]

To split hairs , the "field of real numbers" is technically different than "the vector space of real numbers", but, Platonically, we think of the real numbers as being a single "thing".

I just thought about the mess, if we really distinguished between them: $\mathbb{R}$ as group, ring, algebra, module, field, integral domain, division ring, topological space, manifold, affine space, vector space, Euclidean space, metric space, Hilbert space, Banach space.