Linear algebra MOC

Category of vector spaces

The category of vector spaces 𝖵𝖾𝖼𝗍𝕂 over a field 𝕂 is an example of a concrete category, that is to say its objects are sets with additional structure and its morphisms are mappings that preserve that structure. In this case, each object is a Vector space and each of its morphisms is a Linear map — a mapping which preserves scalar multiplication and vector addition. It is identical to [[Category of left modules|𝕂𝖬𝗈𝖽]], the different name is just for emphasis.

Matrix multiplication algebra as a category.

Universal constructions

• The trivial vector space 0 of dimension zero is both the initial and terminal object i.e. the zero object. See In the Category of vector spaces.

Skeleton

The canonical skeleton category Sk⁡(𝖵𝖾𝖼𝗍𝕂) is the restriction to objects of the form 𝕂(𝛼) for some Cardinal 𝛼. This of course assumes AC.

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