Editing Transformation matrix (section)

==Composing and inverting transformations==

One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily [[composition (functions)|composed]] and inverted.

Composition is accomplished by [[matrix multiplication]]. [[Row and column vectors]] are operated upon by matrices, rows on the left and columns on the right. Since text reads from left to right, column vectors are preferred when transformation matrices are composed:

If '''A''' and '''B''' are the matrices of two linear transformations, then the effect of first applying '''A''' and then '''B''' to a column vector <math>\mathbf{x}</math> is given by:
<math display="block">\mathbf{B}(\mathbf{A} \mathbf x) = (\mathbf{BA}) \mathbf x.</math>

In other words, the matrix of the combined transformation '''''A''' followed by '''B''''' is simply the product of the individual matrices.

When '''A''' is an [[invertible matrix]] there is a matrix '''A'''<sup>−1</sup> that represents a transformation that "undoes" '''A''' since its composition with '''A''' is the [[identity matrix]]. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating in opposite direction) and then composing them in reverse order.  Reflection matrices are a special case because [[Involutory matrix|they are their own inverses]] and don't need to be separately calculated.