Editing Row and column vectors (section)

==Matrix transformations==
{{main|Transformation matrix}}
An {{math|''n'' × ''n''}} matrix {{mvar|M}} can represent a [[linear map]] and act on row and column vectors as the linear map's [[transformation matrix]]. For a row vector {{math|'''v'''}}, the product {{math|'''v'''''M''}} is another row vector {{math|'''p'''}}: 

<math display="block">\mathbf{v} M = \mathbf{p} \,.</math>

Another {{math|''n'' × ''n''}} matrix {{mvar|Q}} can act on {{math|'''p'''}},

<math display="block"> \mathbf{p} Q = \mathbf{t} \,. </math>

Then one can write {{math|1='''t''' = '''p'''''Q'' = '''v'''''MQ''}}, so the [[matrix product]] transformation {{mvar|MQ}} maps {{math|'''v'''}} directly to {{math|'''t'''}}.  Continuing with row vectors, matrix transformations further reconfiguring {{mvar|n}}-space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under an {{math|''n'' × ''n''}} matrix action, the operation occurs to the left,

<math display="block"> \mathbf{p}^\mathrm{T} = M \mathbf{v}^\mathrm{T} \,,\quad \mathbf{t}^\mathrm{T} = Q \mathbf{p}^\mathrm{T},</math>

leading to the algebraic expression {{math|''QM'' '''v'''<sup>T</sup>}}  for the composed output from {{math|'''v'''<sup>T</sup>}} input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.