# Transformation Matrix

## Transformation Matrix

In the Cartesian plane, a transformation matrix is a matrix that uses the coordinates of an initial point represented by a column matrix, to find the coordinates of its image using a geometric transformation. Therefore, the coordinates of the image are obtained by multiplying the matrix of the corresponding geometric transformation by a column vector (the coordinates of a point).

In linear algebra, a geometric transformation in the Cartesian plane may be represented using matrices or vectors. The matrix associated with a transformation can therefore be used to obtain the coordinates of the image of any point of an initial figure by applying the transformation matrix to the coordinates of the point.

### Formula

For all plane geometric transformation, given the point $$P(x, y)$$ and a transformation matrix $$\begin{bmatrix}a & b\\c & d\end{bmatrix}$$, the image $$P'(x’, y’)$$ of point $$P$$ is obtained as follows: $$\begin{bmatrix}a & b\\c & d\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.

### Examples

• For a horizontal scale change of a factor of $$k$$,
the transformation matrix is $$\small{\begin{bmatrix}k & 0\\0 & 1\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}k & 0\\0 & 1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.
• For a translation $$t$$ in a Cartesian plane, defined by a vector $$\overrightarrow{t}(a, b)$$,
the transformation matrix is $$\small{\begin{bmatrix}x + a\\y+b\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}x + a\\y + b\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.
• For a reflection $$s_x$$ with respect to the x-axis n a Cartesian plane,
the transformation matrix is $$\small{\begin{bmatrix}1 & 0\\0 & −1\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}1 & 0\\0 & −1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.
• For a rotation $$r$$ of 90° centred at the origin of a Cartesian plane,
the transformation matrix is $$\small{\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.
• For a rotation $$r$$ of $$\theta°$$ centred at the origin of a Cartesian plane,
the transformation matrix is $$\small{\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.
• For a dilation $$D$$ with a scale factor of $$k$$ centred at the origin of a Cartesian plane,
the transformation matrix is $$\small{\begin{bmatrix}k & 0\\0 & k\end{bmatrix}}$$,
such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ are given by $$\small{\begin{bmatrix}k & 0\\0 & k\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}$$.