Editing Linear combination (section)

== Generalizations==
If ''V'' is a [[topological vector space]], then there may be a way to make sense of certain ''infinite'' linear combinations, using the topology of ''V''.
For example, we might be able to speak of ''a''<sub>1</sub>'''v'''<sub>1</sub>&nbsp;+ ''a''<sub>2</sub>'''v'''<sub>2</sub>&nbsp;+ ''a''<sub>3</sub>'''v'''<sub>3</sub>&nbsp;+&nbsp;⋯, going on forever.
Such infinite linear combinations do not always make sense; we call them ''convergent'' when they do.
Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis.
The articles on the various flavors of topological vector spaces go into more detail about these.

If ''K'' is a [[commutative ring]] instead of a field, then everything that has been said above about linear combinations generalizes to this case without change.
The only difference is that we call spaces like this ''V'' [[module (mathematics)|modules]] instead of vector spaces.
If ''K'' is a [[noncommutative ring]], then the concept still generalizes, with one caveat:
since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module.
This is simply a matter of doing scalar multiplication on the correct side.

A more complicated twist comes when ''V'' is a [[bimodule]] over two rings, ''K''<sub>L</sub> and ''K''<sub>R</sub>.
In that case, the most general linear combination looks like
:<math> a_1 \mathbf v_1 b_1 + \cdots +  a_n \mathbf v_n b_n </math>
where ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> belong to ''K''<sub>L</sub>, ''b''<sub>1</sub>,...,''b''<sub>''n''</sub> belong to ''K''<sub>R</sub>, and '''v'''<sub>1</sub>,…,'''v'''<sub>''n''</sub> belong to ''V''.