Given a field , the category is category whose objects are vector spaces and whose morphisms are linear map.
If the field is understood, one often just writes .
The study of is called linear algebra.
The full subcategory of Vect consisting of finite-dimensional vector spaces is denoted .
is where most of ordinary linear algebra lives, although much of it makes sense in all of . On the other hand, anything involving transposes or inner products really takes place in Hilb.