Transformation matrix

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