See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations

The calculator below will calculate the image of the points in two-dimensional space after applying the transformation.

Transformation Matrices

The above transformations (rotation, reflection, scaling, and shearing) can be represented by matrices. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate.

The transformation matrices are as follows:

Examples:

• Rotate point $A\left(2,3\right)$ clockwise about the origin by an angle $90°$.

  $\begin{array}{rl}\left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{cc}\cos 90°& \sin 90°\\ -\sin 90°& \cos 90°\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}2\\ 3\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{c}3\\ -2\end{array}\right)\end{array}$

  $A^{\prime <!--PRIME-->}\left(3,-2\right)$

• Reflect point $B\left(-3,4\right)$ against the $x$-axis.

  $\begin{array}{rl}\left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}-3\\ 4\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime <!--PRIME-->}\\ y^{\prime <!--PRIME-->}\end{array}\right)& =\left(\begin{array}{c}-3\\ -4\end{array}\right)\end{array}$

  $B^{\prime <!--PRIME-->}\left(-3,-4\right)$

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By Jimmy Sie

See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations