Editing Vector space (section)

===Direct product and direct sum===
{{Main|Direct product|Direct sum of modules}}
The ''direct product'' of vector spaces and the ''direct sum'' of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

The ''direct product''<!--explain direct--> <math>\textstyle{\prod_{i \in I} V_i}</math> of a family of vector spaces <math>V_i</math> consists of the set of all tuples <math>\left(\mathbf{v}_i\right)_{i \in I}</math>, which specify for each index <math>i</math> in some [[index set]] <math>I</math> an element <math>\mathbf{v}_i</math> of <math>V_i</math>.{{sfn|Roman|2005|loc=ch. 1, pp. 31–32}} Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' <math display="inline">\bigoplus_{i \in I} V_i</math> (also called [[coproduct]] and denoted <math display="inline">\coprod_{i \in I}V_i</math>), where only tuples with finitely many nonzero vectors are allowed. If the index set <math>I</math> is finite, the two constructions agree, but in general they are different.