Are Translations Linear Transformations?

Translation is not a linear transformation, but there is a simple and useful trick that allows us to treat it as one (see Exercise 9 below). This geometric point of view is obviously useful when we want to model the motion or changes in shape of an object moving in the plane or in 3-space.

Why is translation not linear transformation?

Translation in an n-dimensional space is not a linear operation but you can make it a linear operation by looking at it from another space. The price of this is adding another dimension. So translation in n dimensions can be expressed as a linear operation in n + 1 n+1 n+1 dimensions.

Is translational operator linear?

Translation operators are linear and unitary. They are closely related to the momentum operator; for example, a translation operator that moves by an infinitesimal amount in the x direction has a simple relationship to the x-component of the momentum operator.