Definition:Transformation Matrix

Definition

A transformation matrix is a matrix defining a coordinate transformation.

Examples

Cartesian Plane

Let $\CC_\alpha$ and $\CC_\beta$ be cartesian coordinate systems over a plane.

Let $\begin {pmatrix} x \\ y \end {pmatrix}$ be the column vector representing an arbitrary point $P$ under $\CC_\alpha$.

Let $\begin {pmatrix} x' \\ y' \end {pmatrix}$ be the column vector representing $P$ in $\CC_\beta$.

Then:

$\begin {pmatrix} x' \\ y' \end {pmatrix} = \mathbf T \begin {pmatrix} x \\ y \end {pmatrix}$

where $\mathbf T$ denotes the transformation matrix of the coordinate transformation from $\CC_\alpha$ to $\CC_\beta$.

Plane Reflection in $x$-Axis

Let $\CC$ be a cartesian plane.

Let $P = \begin {pmatrix} x \\ y \end {pmatrix}$ be the column vector representing an arbitrary point $P$ in $\CC$.

Let $P' = \begin {pmatrix} x' \\ y' \end {pmatrix}$ be the column vector representing $P$ after a reflection in the $x$-axis.

Let $\mathbf T$ denote the transformation matrix to convert $P$ to $P'$.

Then:

$\mathbf T = \begin {pmatrix} 1 & 0 \\ 0 & -1 \end {pmatrix}$

Extension in Direction of $x$-Axis

Let $\CC$ be a cartesian plane.

Let $P = \begin {pmatrix} x \\ y \end {pmatrix}$ be the column vector representing an arbitrary point $P$ in $\CC$.

Let $P' = \begin {pmatrix} x' \\ y' \end {pmatrix}$ be the column vector representing $P$ after an extension in the positive direction of the $x$-axis.

Let $\mathbf T$ denote the transformation matrix to convert $P$ to $P'$.

Then:

$\mathbf T = \begin {pmatrix} k & 0 \\ 0 & 1 \end {pmatrix}$

Also see

• Results about transformation matrices can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): transformation: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): transformation: 2.