Geometrical Meaning of Matrix Multiplication

Matrix multiplication is a versatile tool for many aspects of scientific or technical methods. One particular application of matrix multiplication is the transformation of data in n-dimensional space. Data can be scaled, shifted, rotated, or distorted by a simple matrix multiplication. In order to achieve all these operations by a single transformation matrix, the original data has to be augmented by an additional constant value (preferably 1).

Example: transformation of two-dimensional points. Suppose you have seven data points in two dimensions (x, and y). These seven data points have to be submitted to various transformation operations. Therefore we first augment the data points, denoted by [x_i,y_i], with a constant value, resulting in the point vectors [x_i,y_i,1].

For performing the various transformations, we simply have to adjust the transformation matrix.
 
 

Shift		The coordinates of the data points are shifted by the vector [t1,t2]
Scaling		The points are scaled by the factor s.
Scaling only the y coordinate		Here, only the y coordinates are scaled according to the factor s.
Rotation		A rotation of all points around the origin can be accomplished by using the sines and cosines of the rotation angle (remember the negative sign for the first sine term).