Editing Transformation matrix (section)

==Uses==

Matrices allow arbitrary [[linear transformations]] to be displayed in a consistent format, suitable for computation.<ref name="James_Gentle"></ref>  This also allows transformations to be [[Function composition|composed]] easily (by multiplying their matrices).

Linear transformations are not the only ones that can be represented by matrices.  Some transformations that are non-linear on an n-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> can be represented as linear transformations on the ''n''+1-dimensional space '''R'''<sup>''n''+1</sup>. These include both [[affine transformations]] (such as [[Translation (geometry)|translation]]) and [[projective transformation]]s. For this reason, 4×4 transformation matrices are widely used in [[3D computer graphics]]. These ''n''+1-dimensional transformation matrices are called, depending on their application, ''affine transformation matrices'', ''projective transformation matrices'', or more generally ''non-linear transformation matrices''.  With respect to an ''n''-dimensional matrix, an ''n''+1-dimensional matrix can be described as an [[augmented matrix]].

In the [[physics|physical sciences]], an [[active transformation]] is one which actually changes the physical position of a [[system]], and makes sense even in the absence of a [[coordinate system]] whereas a [[passive transformation]] is a change in the coordinate description of the physical system ([[change of basis]]). The distinction between active and passive [[Transformation (mathematics)|transformations]] is important. By default, by ''transformation'', [[mathematician]]s usually mean active transformations, while [[physicist]]s could mean either.

Put differently, a ''passive'' transformation refers to description of the ''same'' object as viewed from two different coordinate frames.