Editing Row and column spaces (section)

===For matrices over a ring===
Similarly the column space (sometimes disambiguated as ''right'' column space) can be defined for matrices over a [[ring (mathematics)|ring]] {{mvar|K}} as
:<math>\sum\limits_{k=1}^n \mathbf{v}_k c_k</math>
for any {{math|''c''<sub>1</sub>, ..., ''c<sub>n</sub>''}}, with replacement of the vector {{mvar|m}}-space with "[[left and right (algebra)|right]] [[free module]]", which changes the order of [[scalar multiplication]] of the vector {{math|'''v'''<sub>''k''</sub>}} to the scalar {{math|''c<sub>k</sub>''}} such that it is written in an unusual order ''vector''–''scalar''.<ref>Important only if {{mvar|K}} is not [[commutative ring|commutative]]. Actually, this form is merely a [[matrix multiplication|product]] {{math|''A'''''c'''}} of the matrix {{mvar|A}} to the column vector {{math|'''c'''}} from {{math|''K''<sup>''n''</sup>}} where the order of factors is ''preserved'', unlike [[#Definition|the formula above]].</ref>