Category of vector spaces over a field

Fix a field $k$ . The category $k\text{-}\mathbf{Vect}$ has vector spaces over $k$ as objects and $k$ -linear maps as morphisms. Thus a morphism satisfies

f(ax+by)=af(x)+bf(y)\qquad(a,b\in k).

The identity map is linear, and a composite of linear maps is linear; these give the categorical identities and composition.

Scalar field and dimension

The field is fixed throughout a Hom-set. A complex vector space can also be regarded as a real vector space by restriction of scalars, but the allowed maps change. For finite-dimensional spaces of dimensions $m$ and $n$ , bases identify the Hom-space with $M_{n\times m}(k)$ ; the identification depends on the bases.

Units and products

No algebra product or distinguished multiplicative unit is part of this category. Multiplication-preserving or bracket-preserving maps belong to more structured categories.