Transformation Matrix

In linear algebra, linear transformations can be represented by matrices. If T {\displaystyle T} is a linear transformation mapping R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} and x {\displaystyle \mathbf {x} } is a column vector with n {\displaystyle n} entries, then there exists an m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} ,[1] such that: T ( x ) = A x {\displaystyle T(\mathbf {x} )=A\mathbf {x} } Note that A {\displaystyle A} has m {\displaystyle m} rows and n {\displaystyle n} columns, whereas the transformation T {\displaystyle T} is from R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.[2][3]