Rotation and reflection linear transformations

In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping Rⁿ to R^m and ${\displaystyle {\vec {x}}}$ is a column vector with n entries, then

for some m×n matrix A, called the transformation matrix of T. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated 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 Rⁿ can be represented as linear transformations on the n+1-dimensional space Rⁿ⁺¹. These include both affine transformations (such as translation) and projective transformations. 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.

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