Using Transformation Matrices

Understanding Transformation Matrices

A transformation matrix is a special square matrix used to represent linear transformations.
This transformation could be any operation that alters the position, orientation or size of the object, such as translations, rotations and scaling.
Every transformation matrix has a size of 2x2 or 3x3, depending on whether the transformation is in 2D or 3D.
A transformation is applied to a vector or point by multiplying the transformation matrix with the vector.

Identity Matrix: The identity matrix, denoted as I, leaves a vector unchanged when multiplied with it.
Translation Matrix: This matrix is used for translating (moving) a point in 2D or 3D space. Note: Translation cannot be represented using a 2x2 or 3x3 matrix only. An extra dimension must be added for the movement.
Scaling Matrix: This matrix is used to increase or decrease the size of an object. The diagonal elements of a scaling matrix define the scaling factor in each dimension.
Rotation Matrix: A rotation matrix rotates a vector around the origin. The angle of rotation is determined by the values of the matrix.

To apply a transformation to a point or vector, multiply the transformation matrix by the vector. Remember that matrix multiplication is not commutative - the order of multiplication matters.
Combine transformations by multiplying their matrices together. The resulting matrix represents the combined transformation.
When multiple transformations are applied, they are performed in the reverse order of multiplication. This is due to the non-commutative property of matrix multiplication.
Inverse of a transformation matrix undoes the transformation. Multiplying a matrix by its inverse will yield the identity matrix.

Transformation matrices preserve vector addition and scalar multiplication.
The determinant of a rotation matrix is always 1 and its inverse is its transpose.
The determinant of an identity matrix is always 1.
Transformation matrices are linear. This means the transformation of a sum of vectors is the sum of their transformations, and the transformation of a scalar multiplication of a vector is the scalar multiplication of the transformed vector.