Linear Transformations

A linear transformation is an operation that takes a point (x, y) to another point (x', y'). The below simulations applies the given linear transformation to the entire coordinate system within [-5,-5] x [5,5]

Transformation matrix:






Rotate by (in degrees):

Description

A linear transformation L represented by a matrix \( M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) maps x to \( ax + by \) and y to \( cx + dy \).

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} \] Rotation matrix

A matrix that rotates the coordinate system by an angle θ is given by:

\[ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \] Determinant

The determinant of a 2x2 matrix is equal to \( ad - bc \). Figuratively, the determinant determines the scaling of areas that occurs as a result of a linear transformation represented by a matrix. The sign of the determinant is negative if the linear transformation flips the lattice points. A determinant of zero indicates that the linear transformation reduces the coordinate system to a point or a line.

Inverse of a matrix

The inverse \( M^{-1} \) of a matrix \( M \) is said to be a matrix \( M' \) if \( MM' = M'M = I \). When applied as linear transformation, \( M^{-1} \) allows us to reverse the effects of M and vice versa. If the determinant of a matrix is zero, then its inverse does not exist and the linear transformation cannot be reversed. For a 2x2 matrix, \(M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), its inverse is given by:

\[ M^{-1} = \frac{1}{|M|} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix} \]

Clicking on the following button will lead to calculation of inverse of the current matrix and its application.



Note:
  1. The simulation involves heavy computation during transformation and may lag on some devices.
  2. Although not always visible, the coordinate system actually extends to [-10,-10] x [10,10] in order to account for down-scaling
  3. The simulation also depicts the effect of a transformation to the vector [1,0] (drawn in yellow). Efforts are being made to make this vector modifiable.

Developed by ChanRT | Fork me at GitHub