If function transformation was talked about in terms of geometric transformations it may be clearer why functions translate horizontally the way they do. When addressing translations on the Cartesian plane it is natural to introduce translations in this type of notation:translation parallel to the direction of the lines, through a distance ℓ
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
Similarly, the product of translation matrices is given by adding the vectors:
Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).