An affine transformation has the form $T(\boldsymbol{X})=\boldsymbol{A}\boldsymbol{X}+\boldsymbol{t}$. It maps

A region of area $12$ square units is transformed by

Its image has area $30$ square units and its orientation is reversed.

A transformation is defined by

A point is generated iteratively by

An affine transformation is defined by

A transformation is defined by

A design is transformed by first reflecting it in the line $y=x$, then applying a horizontal stretch with scale factor $3$, and finally translating it by $\begin{pmatrix}-2\\5\end{pmatrix}$.

A mapping program transforms the vertices $O(0,0)$, $A(4,0)$ and $B(0,3)$ of a triangular region to $O'(2,-1)$, $A'(8,1)$ and $B'(-1,5)$ respectively. The transformation has the form $T(\boldsymbol{X})=\boldsymbol{M}\boldsymbol{X}+\boldsymbol{t}$.

A robot rotates every point anticlockwise through $\frac{\pi}{2}$ radians about an unknown centre $C$. The point $P(5,2)$ is mapped to $P'(2,5)$.

A logo is first stretched horizontally by scale factor $1.5$ and then reflected in the line through the origin making an angle of $30^\circ$ with the positive $x$-axis.

An affine transformation is defined by $T(\boldsymbol{X})=\begin{pmatrix}2&0.5\\-1&1.5\end{pmatrix}\boldsymbol{X}+\begin{pmatrix}-3\\2\end{pmatrix}$ A triangle is mapped to a triangle with vertices $(-0.5,2.5)$, $(7.5,-1.5)$ and $(1,7)$.

The position of a point is generated by the recurrence $\boldsymbol{X}_{n+1}=\begin{pmatrix}0&-0.6\\0.6&0\end{pmatrix}\boldsymbol{X}_n+\begin{pmatrix}4\\1\end{pmatrix},\qquad \boldsymbol{X}_0=\begin{pmatrix}0\\0\end{pmatrix}$

An iterative design uses the vertices $A(0,0)$, $B(9,0)$, $C(9,9)$ and $D(0,9)$. At each step, the current point is moved one third of the way from a selected vertex towards its current position. Equivalently, for a selected vertex $\boldsymbol{V}$, $T_V(\boldsymbol{X})=\frac13\boldsymbol{X}+\frac23\boldsymbol{V}$

A region of area $24$ square units is transformed by $\boldsymbol{A}=\begin{pmatrix}k&3\\1&k\end{pmatrix}$ Its image has area $60$ square units and its orientation is reversed.

A circular region has centre $(1,-2)$ and radius $3$. It is transformed by $T(\boldsymbol{X})=\begin{pmatrix}1.2&0\\0&0.5\end{pmatrix}\boldsymbol{X}+\begin{pmatrix}4\\1\end{pmatrix}$

The transformation $T$ is a reflection in the line $y=x+2$.

A theatre projects a rectangular image with vertices $O(0,0)$, $P(5,0)$, $Q(5,3)$ and $R(0,3)$. The projector applies the affine transformation

A light on a rotating fairground ride is initially at $X_0=(8,1)$. During each step, the vector from $C$ to the light is multiplied by $0.9$ and rotated anticlockwise through $\frac{\pi}{6}$, where $C=(2,1)$.

A textile motif is processed using a horizontal stretch
$H=\begin{pmatrix}1.5&0\\0&1\end{pmatrix}$
and a reflection $F$ in the line $y=x$. The motif contains the point $P=(2,4)$.

A camera sensor applies a horizontal stretch with scale factor $1.25$, a vertical compression with scale factor $0.8$, and then an anticlockwise rotation through angle $\theta$. A rectangular sensor region has vertices $(0,0)$, $(6,0)$, $(6,4)$ and $(0,4)$.

A digital image occupies the rectangular region $0\leq x\leq800$, $0\leq y\leq600$, where coordinates are measured in pixels. It is transformed by
$T(X)=\begin{pmatrix}0.8&-0.3\\0.2&0.9\end{pmatrix}X+\begin{pmatrix}120\\50\end{pmatrix}$

A transformation is defined by $T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}2&-4\\-1&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}3\\5\end{pmatrix}$

Let $\boldsymbol{H}$ represent a horizontal stretch of scale factor $2$, and let $\boldsymbol{R}$ represent an anticlockwise rotation through $45^\circ$ about the origin.

To align two medical images, a similarity transformation maps $A(1,2)$ to $A'(6,-1)$ and $B(5,2)$ to $B'(6,7)$. The transformation consists of an enlargement, a rotation and a translation.

A point is generated iteratively by $\boldsymbol{X}_{n+1}=-0.5\boldsymbol{X}_n+\begin{pmatrix}6\\-3\end{pmatrix},\qquad \boldsymbol{X}_0=\begin{pmatrix}2\\4\end{pmatrix}$

A digital image is rotated clockwise through $30^\circ$, then stretched horizontally by scale factor $1.4$ and vertically by scale factor $0.8$, and finally translated by $\begin{pmatrix}120\\80\end{pmatrix}$ pixels.

To align two aerial photographs, three landmarks have coordinates $A(1,2)$, $B(5,2)$ and $C(2,6)$ in the first photograph. Their corresponding coordinates in the second photograph are $A'(4,1)$, $B'(10,3)$ and $C'(3,10)$. An affine transformation $T(X)=MX+t$ is used.

A fractal is generated inside the unit square. At each stage, every square is replaced by four copies, each with linear scale factor $0.4$, positioned at the four corners. The lower-left corner transformations include
$T_0(X)=0.4X,\qquad T_1(X)=0.4X+\begin{pmatrix}0.6\\0\end{pmatrix}$
The other two transformations are denoted by $T_2$ and $T_3$.

A logo is transformed by the parameter-dependent matrix
$A_k=\begin{pmatrix}k&2\\1&k-1\end{pmatrix}$
followed by a translation through $(4,-3)$. The original logo has area $20$ square units.

A robotic camera repeatedly corrects the position of a target using