Transformation Matrix ○꠹ Entries

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Transformation Matrix ○꠹｜Definition｜1st｜20251119205401-00-⌔

Transformation matrix - Wikipedia

Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If $T$ is a linear transformation mapping ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ and $\mathbf{x}$ is a column vector with $n$ entries, then there exists an $m\times n$ matrix $A$, called the transformation matrix of $T$,¹ such that:

$T(\mathbf{x})=A\mathbf{x}$

Note that $A$ has $m$ rows and $n$ columns, whereas the transformation $T$ is from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.²³

Printed 2026-06-28.

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Footnotes

1. Gentle, James E. (2007). “Matrix Transformations and Factorizations”. Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer. ISBN 9780387708737. ↩

2. Rafael Artzy (1965) Linear Geometry ↩

3. J. W. P. Hirschfeld (1979) Projective Geometry of Finite Fields, Clarendon Press ↩

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