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 \(\small{\begin{bmatrix}k & 0\\0 & k\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}}\).