Editing Transformation matrix (section)

{{Short description|Central object in linear algebra; mapping vectors to vectors}}
{{Use American English|date=January 2019}}
In [[linear algebra]], [[linear transformation]]s can be represented by [[matrix (mathematics)|matrices]]. If <math>T</math> is a linear transformation mapping <math>\mathbb{R}^n</math> to <math>\mathbb{R}^m</math> and <math>\mathbf x</math> is a [[column vector]] with <math>n</math> entries, then there exists an <math>m \times n</math> matrix <math>A</math>, called the '''transformation matrix''' of <math>T</math>,<ref name="James_Gentle">{{cite book | last = Gentle | first = James E. |chapter = Matrix Transformations and Factorizations |title = Matrix Algebra: Theory, Computations, and Applications in Statistics |publisher = Springer |year = 2007 |isbn = 9780387708737 |chapter-url = https://books.google.com/books?id=PDjIV0iWa2cC&pg=PA172 }}</ref> such that:
<math display="block">T( \mathbf x ) = A \mathbf x</math>
Note that <math>A</math> has <math>m</math> rows and <math>n</math> columns, whereas the transformation <math>T</math> is from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^m</math>.  There are alternative expressions of transformation matrices involving [[row vector]]s that are preferred by some authors.<ref>[[Rafael Artzy]] (1965) ''Linear Geometry''</ref><ref>[[J. W. P. Hirschfeld]] (1979) ''Projective Geometry of Finite Fields'', [[Clarendon Press]]</ref>