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Definition

Given a field $k$, the category ${\mathrm{Vect}}_k$ is category whose objects are vector spaces and whose morphisms are linear map.

If the field $k$ is understood, one often just writes $\mathrm{Vect}$.

The study of $\mathrm{Vect}$ is called linear algebra.

Related categories

The full subcategory of Vect consisting of finite-dimensional vector spaces is denoted $Fin\mathrm{Vect}$.

$Fin\mathrm{Vect}$ is where most of ordinary linear algebra lives, although much of it makes sense in all of $\mathrm{Vect}$. On the other hand, anything involving transposes or inner products really takes place in $\mathrm{Fin}$ Hilb.